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17.E: Second-Order Differential Equations (Exercises) - Mathematics


17.1: Second-Order Linear Equations

Q17.1.1

Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous.

  1. (x^3y''+(x-1)y'-8y=0)
  2. ((1+y^2)y''+xy'-3y= cos x)
  3. (xy''+e^yy'=x)
  4. (y''+ frac{4}{x}y'-8xy=5x^2+1 )
  5. (y''+( sin x)y'-xy=4y )
  6. (y''+(frac{x+3}{y})y'=0)

Q17.1.2

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c1 and c2. What do the solutions have in common?

  1. [T] (y''+2y'-3y=0; ~~ y(x)=c_1e^x+c_2e^{-3x})
  2. [T](x^2y''-2y-3x^2+1=0; ~~y(x)=c_1x^2+c_2x^{-1}+x^2 ln(x)+ frac{1}{2} )
  3. [T] (y''+14y+49y=0;y″+14y′+49y=0; y(x)=c1e−7x+c2xe−7xy(x)=c1e−7x+c2xe−7x
  4. [T]6y''−49y′+8y=0;6y″−49y′+8y=0; y(x)=c1ex/6+c2e8xy(x)=c1ex/6+c2e8x

Q17.1.3

Find the general solution to the linear differential equation.

y''−3y′−10y=0y″−3y′−10y=0

y''−7y′+12y=0y″−7y′+12y=0

y''+4y′+4y=0y″+4y′+4y=0

4y''−12y′+9y=04y″−12y′+9y=0

2y''−3y′−5y=02y″−3y′−5y=0

3y''−14y′+8y=03y″−14y′+8y=0

y''+y′+y=0y″+y′+y=0

5y''+2y′+4y=05y″+2y′+4y=0

y''−121y=0y″−121y=0

8y''+14y′−15y=08y″+14y′−15y=0

y''+81y=0y″+81y=0

y''−y′+11y=0y″−y′+11y=0

2y''=02y″=0

y''−6y′+9y=0y″−6y′+9y=0

3y''−2y′−7y=03y″−2y′−7y=0

4y''−10y′=04y″−10y′=0

36d2ydx2+12dydx+y=036d2ydx2+12dydx+y=0

25d2ydx2−80dydx+64y=025d2ydx2−80dydx+64y=0

d2ydx2−9dydx=0d2ydx2−9dydx=0

4d2ydx2+8y=04d2ydx2+8y=0

Q17.1.4

Solve the initial-value problem.

y''+5y′+6y=0,y(0)=0,y′(0)=−2y″+5y′+6y=0,y(0)=0,y′(0)=−2

y''+2y′−8y=0,y(0)=5,y′(0)=4y″+2y′−8y=0,y(0)=5,y′(0)=4

y''+4y=0,y(0)=3,y′(0)=10y″+4y=0,y(0)=3,y′(0)=10

y''−18y′+81y=0,y(0)=1,y′(0)=5y″−18y′+81y=0,y(0)=1,y′(0)=5

y''−y′−30y=0,y(0)=1,y′(0)=−16y″−y′−30y=0,y(0)=1,y′(0)=−16

4y''+4y′−8y=0,y(0)=2,y′(0)=14y″+4y′−8y=0,y(0)=2,y′(0)=1

25y''+10y′+y=0,y(0)=2,y′(0)=125y″+10y′+y=0,y(0)=2,y′(0)=1

y''+y=0,y(π)=1,y′(π)=−5y″+y=0,y(π)=1,y′(π)=−5

Solve the boundary-value problem, if possible.

y''+y′−42y=0,y(0)=0,y(1)=2y″+y′−42y=0,y(0)=0,y(1)=2

9y''+y=0,y(3π2)=6,y(0)=−89y″+y=0,y(3π2)=6,y(0)=−8

y''+10y′+34y=0,y(0)=6,y(π)=2y″+10y′+34y=0,y(0)=6,y(π)=2

y''+7y′−60y=0,y(0)=4,y(2)=0y″+7y′−60y=0,y(0)=4,y(2)=0

y''−4y′+4y=0,y(0)=2,y(1)=−1y″−4y′+4y=0,y(0)=2,y(1)=−1

y''−5y′=0,y(0)=3,y(−1)=2y″−5y′=0,y(0)=3,y(−1)=2

y''+9y=0,y(0)=4,y(π3)=−4y″+9y=0,y(0)=4,y(π3)=−4

4y''+25y=0,y(0)=2,y(2π)=−24y″+25y=0,y(0)=2,y(2π)=−2

Find a differential equation with a general solution that is y=c1ex/5+c2e−4x.y=c1ex/5+c2e−4x.

Q17.1.X

Find a differential equation with a general solution that is y=c1ex+c2e−4x/3.y=c1ex+c2e−4x/3.

For each of the following differential equations:

  1. Solve the initial value problem.
  2. [T] Use a graphing utility to graph the particular solution.

y''+64y=0;y(0)=3,y′(0)=16y″+64y=0;y(0)=3,y′(0)=16

y''−2y′+10y=0y(0)=1,y′(0)=13y″−2y′+10y=0y(0)=1,y′(0)=13

y''+5y′+15y=0y(0)=−2,y′(0)=7y″+5y′+15y=0y(0)=−2,y′(0)=7

Q17.1.X

(Principle of superposition) Prove that if y1(x)y1(x) and y2(x)y2(x) are solutions to a linear homogeneous differential equation, y''+p(x)y′+q(x)y=0,y″+p(x)y′+q(x)y=0, then the function y(x)=c1y1(x)+c2y2(x),y(x)=c1y1(x)+c2y2(x), where c1c1 and c2c2 are constants, is also a solution.

Prove that if a, b, and c are positive constants, then all solutions to the second-order linear differential equation ay''+by′+cy=0ay″+by′+cy=0 approach zero as x→∞.x→∞. (Hint: Consider three cases: two distinct roots, repeated real roots, and complex conjugate roots.)

17.2: Nonhomogeneous Linear Equations

Solve the following equations using the method of undetermined coefficients.

(2y''−5y′−12y=6)

(3y''+y′−4y=8)

(y=c_1e^{−4x/3}+c_2e^x−2)

(y''−6y′+5y=e^{−x})

(y''+16y=e^{−2x})

(y=c_1 cos4x+c_2 sin 4x+frac{1}{20}e^{−2x})

(y″−4y=x^2+1)

(y″−4y′+4y=8x^2+4x)

(y=c_1e^{2x}+c_2xe^{2x}+2x^2+5x)

(y″−2y′−3y= sin 2x)

(y″+2y′+y= sin x+ cos x)

(y=c_1e^{−x}+c_2xe^{−x}+frac{1}{2} sin x−frac{1}{2} cos x)

(y″+9y=e^x cos x)

(y″+y=3 sin 2x+x cos 2x)

(y=c_1 cos x+ c_2 sin x−frac{1}{3}x cos 2x−frac{5}{9} sin 2x)

(y″+3y′−28y=10e{4x})

(y″+10y′+25y=xe^{−5x}+4)

(y=c_1e^{−5x}+c_2xe^{−5x}+frac{1}{6}x^3e^{−5x}+frac{4}{25})

In each of the following problems,

  1. Write the form for the particular solution (y_p(x)) for the method of undetermined coefficients.
  2. [T] Use a computer algebra system to find a particular solution to the given equation.

(y″−y′−y=x+e^{−x})

(y″−3y=x^2−4x+11)

a. (y_p(x)=Ax^2+Bx+C)

b. (y_p(x)=−frac{1}{3}x^2+frac{4}{3}x−frac{35}{9})

(y''−y′−4y=e^x cos 3x )

(2y″−y′+y=(x^2−5x)e^{−x})

a. (y_p(x)=(Ax^2+Bx+C)e^{−x})

b. (y_p(x)=(frac{1}{4}x^2−frac{5}{8}x−frac{33}{32})e^{−x})

(4y″+5y′−2y=e^{2x}+x sin x)

(y''−y′−2y=x^2e^x sin x)

a. (y_p(x)=(Ax^2+Bx+C)e^x cos x+(Dx^2+Ex+F)e^x sin x)

b. (y_p(x)=(−frac{1}{10}x^2−frac{11}{25}x−frac{27}{250})e^x cos x +(−frac{3}{10}x^2+frac{2}{25}x+frac{39}{250})e^x sin x)

Solve the differential equation using either the method of undetermined coefficients or the variation of parameters.

(y″+3y′−4y=2e^x)

(y''+2y′=e^{3x})

(y=c_1+c_2e^{−2x}+frac{1}{15}e^{3x})

(y''+6y′+9y=e^{−x})

(y''+2y′−8y=6e^{2x})

(y=c_1e^{2x}+c_2e^{−4x}+xe^{2x})

Solve the differential equation using the method of variation of parameters.

(4y″+y=2 sin x)

(y″−9y=8x)

(y=c_1e^{3x}+c_2e^{−3x}−frac{8x}{9})

(y″+y= sec x, ;;;;;;;;;;0

(y″+4y=3 csc 2x,;;;;;;;;;; 0

(y=c_1 cos 2x+c_2 sin 2x−frac{3}{2} x cos 2x+frac{3}{4} sin 2x ln ( sin 2x))

Find the unique solution satisfying the differential equation and the initial conditions given, where (y_p(x)) is the particular solution.

(y″−2y′+y=12e^x,y_p(x)=6x^2e^x, y(0)=6,y′(0)=0)

(y''−7y′=4xe^{7x},y_p(x)=frac{2}{7}x^2e^{7x}−frac{4}{49}xe^{7x},y(0)=−1,y'(0)=0)

(y=− frac {347}{343}+ frac {4}{343}e^{7x}+frac{2}{7}x^2e^{7x}−frac{4}{49}xe^{7x})

(y″+y= cos x−4 sin x, y_p(x)=2x cos x+frac{1}{2} x sin x, y(0)=8,y′(0)=−4)

(y″−5y′=e^{5x}+8e^{−5x}, y_p(x)=frac{1}{5}xe^{5x}+frac{4}{25}e^{−5x}, y(0)=−2,y′(0)=0)

(y=−frac{57}{25}+frac{3}{25}e^{5x}+frac{1}{5}xe^{5x}+frac{4}{25}e^{−5x})

In each of the following problems, two linearly independent solutions—(y_1) and (y_2)—are given that satisfy the corresponding homogeneous equation. Use the method of variation of parameters to find a particular solution to the given nonhomogeneous equation. Assume x > 0 in each exercise.

(x^2y″+2xy′−2y=3x, y_1(x)=x,y2(x)=x^{−2})

(x^2y''−2y=10x^2−1,y_1(x)=x^2,y_2(x)=x^{−1})

(y_p=frac{1}{2}+frac{10}{3}x^2 ln x)

17.3: Applications

A mass weighing 4 lb stretches a spring 8 in. Find the equation of motion if the spring is released from the equilibrium position with a downward velocity of 12 ft/sec. What is the period and frequency of the motion?

A mass weighing 2 lb stretches a spring 2 ft. Find the equation of motion if the spring is released from 2 in. below the equilibrium position with an upward velocity of 8 ft/sec. What is the period and frequency of the motion?

(x″+16x=0, x(t)=frac{1}{6} cos (4t)−2 sin (4t),) period (=frac{π}{2} ext{sec},) frequency (=frac{2}{π} ext{Hz})

A 100-g mass stretches a spring 0.1 m. Find the equation of motion of the mass if it is released from rest from a position 20 cm below the equilibrium position. What is the frequency of this motion?

A 400-g mass stretches a spring 5 cm. Find the equation of motion of the mass if it is released from rest from a position 15 cm below the equilibrium position. What is the frequency of this motion?

(x″+196x=0, x(t)=0.15 cos (14t),) period (=frac{π}{7} ext{sec},) frequency (=frac{7}{π} ext{Hz})

A block has a mass of 9 kg and is attached to a vertical spring with a spring constant of 0.25 N/m. The block is stretched 0.75 m below its equilibrium position and released.

  1. Find the position function (x(t)) of the block.
  2. Find the period and frequency of the vibration.
  3. Sketch a graph of (x(t)).
  4. At what time does the block first pass through the equilibrium position?

A block has a mass of 5 kg and is attached to a vertical spring with a spring constant of 20 N/m. The block is released from the equilibrium position with a downward velocity of 10 m/sec.

  1. Find the position function (x(t)) of the block.
  2. Find the period and frequency of the vibration.
  3. Sketch a graph of (x(t)).
  4. At what time does the block first pass through the equilibrium position?

a. (x(t)=5 sin (2t))

b. period (=π ext{sec},) frequency (=frac{1}{π} ext{Hz}) c. d. (t=frac{π}{2} ext{sec})

A 1-kg mass is attached to a vertical spring with a spring constant of 21 N/m. The resistance in the spring-mass system is equal to 10 times the instantaneous velocity of the mass.

  1. Find the equation of motion if the mass is released from a position 2 m below its equilibrium position with a downward velocity of 2 m/sec.
  2. Graph the solution and determine whether the motion is overdamped, critically damped, or underdamped.

An 800-lb weight (25 slugs) is attached to a vertical spring with a spring constant of 226 lb/ft. The system is immersed in a medium that imparts a damping force equal to 10 times the instantaneous velocity of the mass.

  1. Find the equation of motion if it is released from a position 20 ft below its equilibrium position with a downward velocity of 41 ft/sec.
  2. Graph the solution and determine whether the motion is overdamped, critically damped, or underdamped.

a. (x(t)=e^{−t/5}(20 cos (3t)+15 sin(3t)))

b. underdamped

A 9-kg mass is attached to a vertical spring with a spring constant of 16 N/m. The system is immersed in a medium that imparts a damping force equal to 24 times the instantaneous velocity of the mass.

  1. Find the equation of motion if it is released from its equilibrium position with an upward velocity of 4 m/sec.
  2. Graph the solution and determine whether the motion is overdamped, critically damped, or underdamped.

A 1-kg mass stretches a spring 6.25 cm. The resistance in the spring-mass system is equal to eight times the instantaneous velocity of the mass.

  1. Find the equation of motion if the mass is released from a position 5 m below its equilibrium position with an upward velocity of 10 m/sec.
  2. Determine whether the motion is overdamped, critically damped, or underdamped.

a. (x(t)=5e^{−4t}+10te^{−4t})

b. critically damped

A 32-lb weight (1 slug) stretches a vertical spring 4 in. The resistance in the spring-mass system is equal to four times the instantaneous velocity of the mass.

  1. Find the equation of motion if it is released from its equilibrium position with a downward velocity of 12 ft/sec.
  2. Determine whether the motion is overdamped, critically damped, or underdamped.

A 64-lb weight is attached to a vertical spring with a spring constant of 4.625 lb/ft. The resistance in the spring-mass system is equal to the instantaneous velocity. The weight is set in motion from a position 1 ft below its equilibrium position with an upward velocity of 2 ft/sec. Is the mass above or below the equation position at the end of (π) sec? By what distance?

(x(π)=frac{7e^{−π/4}}{6}) ft below

A mass that weighs 8 lb stretches a spring 6 inches. The system is acted on by an external force of (8 sin 8t )lb. If the mass is pulled down 3 inches and then released, determine the position of the mass at any time.

A mass that weighs 6 lb stretches a spring 3 in. The system is acted on by an external force of (8 sin (4t) ) lb. If the mass is pulled down 1 inch and then released, determine the position of the mass at any time.

(x(t)=frac{32}{9} sin (4t)+ cos (sqrt{128}t)−frac{16}{9sqrt{2}} sin (sqrt{128}t))

Find the charge on the capacitor in an RLC series circuit where (L=40) H, (R=30Ω), (C=1/200) F, and (E(t)=200) V. Assume the initial charge on the capacitor is 7 C and the initial current is 0 A.

Find the charge on the capacitor in an RLC series circuit where (L=2) H, (R=24Ω,) (C=0.005) F, and (E(t)=12 sin 10t) V. Assume the initial charge on the capacitor is 0.001 C and the initial current is 0 A.

(q(t)=e^{−6t}(0.051 cos (8t)+0.03825 sin (8t))−frac{1}{20} cos (10t))

A series circuit consists of a device where(L=1) H, (R=20Ω,) (C=0.002) F, and (E(t)=12) V. If the initial charge and current are both zero, find the charge and current at time t.

A series circuit consists of a device where (L=12) H, (R=10Ω), (C=frac{1}{50}) F, and (E(t)=250) V. If the initial charge on the capacitor is 0 C and the initial current is 18 A, find the charge and current at time t.

(q(t)=e^{−10t}(−32t−5)+5,I(t)=2e^{−10t}(160t+9))

17.4: Series Solutions of Differential Equations

Find a power series solution for the following differential equations.

(y″+6y′=0)

(5y″+y′=0)

(y=c_0+5c_1 sum_{n=1}^∞ frac{(−x/5)^n}{n!}=c_0+5c_1e^{−x/5})

(y''+25y=0)

(y''−y=0)

(y=c_0 sum_{n=0}^∞ frac{(x)^{2n}}{(2n)!}+c_1 sum_{n=0}^∞ frac{(x)^{2n+1}}{(2n+1)!})

(2y′+y=0}

(y′−2xy=0)

(y=c_0 sum_{n=0}^∞ frac{x^{2n}}{n!}=c_0e^{x2})

((x−7)y′+2y=0)

(y''−xy′−y=0)

(y=c_0 sum_{n=0}^∞ frac{x^{2n}}{2^nn!}+c_1 sum_{n=0}^∞ frac{x^{2n+1}}{1⋅3⋅5⋅7⋯(2n+1)})

((1+x^2)y''−4xy′+6y=0)

(x^2y''−xy′−3y=0)

(y=c_1x^3+frac{c_2}{x})

(y″−8y′=0, ; ; ; ; ; ; y(0)=−2,y′(0)=10)

(y″−2xy=0, ; ; ; ; ; ; y(0)=1,y′(0)=−3)

(y=1−3x+frac{2x^3}{3!}−frac{12x^4}{4!}+frac{16x^6}{6!}−frac{120x^7}{7!}+⋯)

The differential equation (x^2y″+xy′+(x^2−1)y=0) is a Bessel equation of order 1. Use a power series of the form (y=sum_{n=0}^∞ a_nx^n) to find the solution.

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

If (y) and (z) are both solutions to (y''+2y′+y=0,) then (y+z) is also a solution.

True

The following system of algebraic equations has a unique solution:

(egin{align} 6z1+3z2 =8 4z1+2z2 =4. end{align})

(y=e^x cos (3x)+e^x sin (2x)) is a solution to the second-order differential equation (y″+2y′+10=0.)

False

To find the particular solution to a second-order differential equation, you need one initial condition.

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation.

(y″−2y=0)

second order, linear, homogeneous, (λ^2−2=0)

(y''−3y+2y= cos (t))

((frac{dy}{dt})^2+yy′=1)

first order, nonlinear, nonhomogeneous

(frac{d^2y}{dt^2}+t frac{dy}{dt}+sin^2 (t)y=e^t)

For the following problems, find the general solution.

(y''+9y=0)

(y=c_1 sin (3x)+c_2 cos (3x))

(y''+2y′+y=0)

(y''−2y′+10y=4x)

(y=c_1e^x sin (3x)+c_2e^x cos (3x)+frac{2}{5}x+frac{2}{25})

(y''= cos (x)+2y′+y)

(y''+5y+y=x+e^{2x})

(y=c_1e^{−x}+c_2e^{−4x}+frac{x}{4}+frac{e^{2x}}{18}−frac{5}{16})

(y''=3y′+xe^{−x})

(y''−x^2=−3y′−frac{9}{4}y+3x)

(y=c_1e^{(−3/2)x}+c_2xe^{(−3/2)x}+frac{4}{9}x^2+frac{4}{27}x−frac{16}{27})

(y''=2 cos x+y′−y)

For the following problems, find the solution to the initial-value problem, if possible.

(y''+4y′+6y=0, y(0)=0, y′(0)=sqrt{2})

(y=e^{−2x} sin (sqrt{2}x))

(y''=3y− cos (x), y(0)=frac{9}{4}, y′(0)=0)

For the following problems, find the solution to the boundary-value problem.

(4y′=−6y+2y″, y(0)=0, y(1)=1)

(y=frac{e^{1−x}}{e^4−1}(e^{4x}−1))

(y''=3x−y−y′, y(0)=−3, y(1)=0)

For the following problem, set up and solve the differential equation.

The motion of a swinging pendulum for small angles (θ) can be approximated by (frac{d^2θ}{dt^2}+frac{g}{L}θ=0,) where (θ) is the angle the pendulum makes with respect to a vertical line, g is the acceleration resulting from gravity, and L is the length of the pendulum. Find the equation describing the angle of the pendulum at time (t,) assuming an initial displacement of (θ_0) and an initial velocity of zero.

(θ(t)=θ_0 cos (sqrt{frac{g}{l}}t))

The following problems consider the “beats” that occur when the forcing term of a differential equation causes “slow” and “fast” amplitudes. Consider the general differential equation (ay″+by= cos (ωt)) that governs undamped motion. Assume that (sqrt{frac{b}{a}}≠ω.)

Find the general solution to this equation (Hint: call (ω_0=sqrt{b/a})).

Assuming the system starts from rest, show that the particular solution can be written as(y=frac{2}{a(ω_0^2−ω^2)} sin (frac{ω_0−ωt}{2}) sin(frac{ω_0+ωt}{2}).)

[T] Using your solutions derived earlier, plot the solution to the system (2y″+9y= cos (2t)) over the interval (t=[−50,50].) Find, analytically, the period of the fast and slow amplitudes.

For the following problem, set up and solve the differential equations.

An opera singer is attempting to shatter a glass by singing a particular note. The vibrations of the glass can be modeled by (y″+ay= cos (bt)), where (y''+ay=0) represents the natural frequency of the glass and the singer is forcing the vibrations at ( cos (bt)).For what value bb would the singer be able to break that glass? (Note: in order for the glass to break, the oscillations would need to get higher and higher.)

(b=sqrt{a})


By multiplying both terms by $y'$ and integrating from $ to $x$ we get: $y'^2 = C + y^2 - frac<1><2>y^4 ag<1>$ as you stated in the comments. If $C eq 0$ and $y(0) eq 0$, a qualitative study of $(1)$ gives that the solutions approach the line $y=y_0$, with $frac<1><2>y_0^4-y_0^2 = C$. In order to have $fin L^2$, $C$ must be zero. Given that: $ g(t)=intfrac

>=logleft(frac<2+sqrt<4-2t^2>> ight)$ we have $ g(y(x)) = x+c $ hence: $frac<2+sqrt<4-2y^2>>=K e^ ag<2>$ with $K=frac<2+sqrt<4-2y(0)^2>>$. From $(2)$ it follows that $|y|leqsqrt<2>$ and : $ y(x) = frac<4K e^x><1+2K^2 e^<2x>>, ag<3>$ hence the sign of $y(x)$ is always the same as the sign of $y(0)$, and by assuming $y(0)>0$, $y(x)$ reaches its maximum, $sqrt<2>$, in $x=-log(xsqrt<2>)$. Here there are the solutions for $y(0)=frac<1><4>,frac<1><8>,frac<1><16>$:

$hspace1in$


Exercises 17.7

Find the general solution to the differential equation using variation of parameters.

Ex 17.7.1 $dsddot y+y= an x$ (answer)

Ex 17.7.3 $dsddot y+4y=sec x$ (answer)

Ex 17.7.4 $dsddot y+4y= an x$ (answer)

Ex 17.7.5 $dsddot y+dot y-6y=t^2e^<2t>$ (answer)

Ex 17.7.6 $dsddot y-2dot y+2y=e^ an(t)$ (answer)

Ex 17.7.7 $dsddot y-2dot y+2y=sin(t)cos(t)$ (This is rather messy when done by variation of parameters compare to undetermined coefficients.) (answer)


Honors Mathematics IV Ordinary Differential Equations

We will also use excerpts of more diverse books for various parts of the course:

Background and Goals: The sequence Honors Math Vv186-285-286 is an introduction to calculus at the honors level. It differs from the Applied Calculus sequence in that new concepts are often introduced in an abstract context, so that they can be applied in more general settings later. Most theorems are proven and new ideas are shown to evolve from previously established theory. Initially, there are fewer applications, as the emphasis is on first establishing a solid mathematical background before proceeding to the analysis of complex models.

The present course focuses on ordinary differential equations and their applications.

Key Words: Ordinary differential equations (ODEs) of first order systems of first-order equations the existence and uniqueness theorem of Picard-Lindeloef eigenvalue problems, diagonalization and the spectral theorem Jordan normal form application to linear systems of first-order equations linear second-order equations elements of complex analysis and residue theory the Laplace transform and its inverse with applications to ODEs power series solutions of ODEs by the Frobenius method Bessel’s and Legendre’s differential equations the Weierstrass approximation theorem and generalized Fourier series introduction to the classical partial differential equations of physics and some basic solutions by separation of variables.

Detailed Content: This course consists of four distinct parts. In the first part, we will discuss some basic integrable single first order ordinary differential equations. In particular, we will look at several types of explicit and implicit equations, including homogeneous, separable, linear, Bernoulli, Ricatti, Clairaut and d'Alembert equations. We will also look at some concrete modeling examples, such as C-14 dating (using the differential equation for unrestricted growth or decay to zero) and population models (using various flavors of the logistic equation).

In the second part, we will discuss systems of first order equations. After proving a general existence and uniqueness theorem (which also has practical applications) we will introduce some background in linear algebra, in particular eigenvalue problems and matrix similarity. This part rounds off the linear algebra that was treated in the Vv285 course and thus completes our "embedded" linear algebra course. Using these techniques, we will be able to solve constant-coefficient linear systems exactly. Next, we will give a brief introduction to general systems of equations, which touches upon the theory of dynamical systems.

The third part is devoted to integration techniques for solving second-order differential equations, with an emphasis on the Laplace transform. In order to get a full grasp of the inverse transform, it is necessary to learn about residue calculus in elementary complex analysis. Since this is also useful elsewhere, and the general concepts of complex analysis will pop up again in more advanced courses, we will devote several lectures to an introduction to complex analysis. Following this, we are able to introduce the Heaviside operator calculus for solving differential equations and from that deduce the Laplace transform technique. Since the Dirac "delta function" is used frequently in applications, we will, moreover, give a brief introduction to locally convex spaces and the space of tempered distributions as dual to the Schwartz space of functions of rapid decrease.

In the last part of the course we discuss series-based solutions. the power-series-based Frobenius method leads us to the Bessel functions, which turn out to have a wide range of applications in physics and engineering. We discuss the problem of a hanging chain, self-buckling of a column, diffraction by a circular aperture and more. Series solutions based on trigonometric functions lead to Fourier series, which we view in the general context of orthogonal functions. We also apply this theory to orthogonal Bessel functions and Legendre polynomials and use these to treat some classical partial differential equations by separation of variables.

Alternatives: Vv256 (Applied Calculus IV) is an applications-oriented course, which covers much of the same material.

[B] Braun, M., Differential Equations and their Applications

[W] Walter, W., Ordinary Differential Equations

[J] Jänich, K., Linear Algebra

[S] Stein, E. M. and Shakarchi, R., Complex Analysis

Lecture Lecture Subject Textbook
1 Introduction and Explicit First-Order ODEs [W] Ch. 1, § 1
2 Separable Equations [W] Ch. 1, § 2
3 Linear and Transformable Equations [W] Ch. 1, § 2
4 Integral Curves and Implicit Equations [W] Ch. 1, §§ 3,4
5 Systems of First-Order ODEs [W] Ch. 3, § 10
6 The Eigenvalue Problem [J] Ch. 9
7 The Spectral Theorem for Self-Adjoint Matrices [J] Ch. 10
8 The Jordan Normal Form [J] Ch. 11.3
9 Linear Systems of First-Order ODEs [B] Sec. 3.11, 3.12
10 Vibrations [B] Sec. 2.6
11 First Midterm Exam
12 Complex Analysis [S] Ch. 1
13 Properties of Holomorphic Functions [S] Ch. 2
14 Singularities and Poles [S] Ch. 3
15 Residue Calculus [S] Ch. 3
16 The Heaviside Operator Method --
17 The Laplace Transform [B] Sec. 2.9-2.13
18 The Laplace Transform [B] Sec. 2.9-2.13
19 The Fourier Transform [S] Ch. 4
20 Second Midterm Exam
21 Power Series Solutions to Second Order ODEs [B] Sec. 2.8
22 Power Series Solutions to Second Order ODEs [B] Sec. 2.8
23 Applications of Bessel Functions [K]
24 Applications of Bessel Functions [K]
25 Orthonormal Functions [B] Ch. 5
26 Fourier Series [B] Ch. 5
27 Boundary Value Problems [B] Ch. 5
28 The Wave and Heat Equations [B] Ch. 5
29 The Wave and Heat Equations [B] Ch. 5
30 Final Exam

(There will always be minor modifications from one iteration of the course to the next if you are presently taking the course, it is not advisable to print out all these slides at the beginning of the term.)


Second Order Differential Equations

Second Order Differential Equations presents a classical piece of theory concerning hypergeometric special functions as solutions of second-order linear differential equations. The theory is presented in an entirely self-contained way, starting with an introduction of the solution of the second-order differential equations and then focusingon the systematic treatment and classification of these solutions.

Each chapter contains a set of problems which help reinforce the theory. Some of the preliminaries are covered in appendices at the end of the book, one of which provides an introduction to Poincaré-Perron theory, and the appendix also contains a new way of analyzing the asymptomatic behavior of solutions of differential equations.

This textbook is appropriate for advanced undergraduate and graduate students in Mathematics, Physics, and Engineering interested in Ordinary and Partial Differntial Equations. A solutions manual is available online.

Gerhard Kristensson received his B.S. degree in mathematics and physics in 1973, and the Ph.D. degree in theoretical physics in 1979, both from the University of Göteborg, Sweden. In 1983 he was appointed Docent in theoretical physics at the University of Göteborg. During 1977-1984 he held a research position sponsored by the National Swedish Board for Technical Development (STU) and he was Lecturer at the Institute of Theoretical Physics, Göteborg from 1980-1984. In 1984-1986 he was a Visiting Scientist at the Applied Mathematical Sciences group, Ames Laboratory, Iowa State University. He held a Docent position at the Department of Electromagnetic Theory, Royal Institute of Technology, Stockholm during 1986-1989, and in 1989 he was appointed the Chair of Electromagnetic Theory at Lund Institute of Technology, Sweden. In 1992, 1997 and 2007 he was a Visiting Erskine Fellow at the Department of Mathematics, University of Canterbury, Christchurch, New Zealand. Currently, Gerhard Kristensson is a member of the Advisory Board of Inverse Problems, the Board of Editors of Wave Motion, and the Editorial and Review Board of Journal of Electromagnetic Waves and Applications and Progress in Electromagnetic Research. He is a Fellow of the Institute of Physics, and since 2006 he is the chairman of the Swedish National committee of Radio Science (SNRV) and official representative for Sweden in the International Union of Radio Science (URSI). From 1994-2005, he was the chairman of Commission B of SNRV and Official Member of URSI, Commission B for Sweden. Kristensson's major research interests are focused on wave propagation in inhomogeneous media, especially inverse scattering problems. During recent years the propagation of transient electromagnetic waves in complex media, such as dispersive anisotropic and bi-isotropic media, has been stressed. High frequency scattering methods, asymptotic expansions, optical fibers, antenna problems, and mixture formulas are also of interest, as well as radome design problems and homogenization of complex materials.

“Gerhard Kristensson’s book Second Order Differential Equations: Special Functions and Their Classification concerns precisely what Felix Klein called ‘the central problem of the whole of modern [i.e. late nineteenth century] mathematics.’ … The book is well-written … . A big picture of special function relationships emerges by the end and the book has several helpful diagrams to help visualize these relationships. … An instructor teaching from this book might do well … .” (John D. Cook, The Mathematical Association of America, October, 2010)

“The aim of the author is to present the classification and systematics of the pertinent special functions on the basis of the solutions of the underlying differential equations. … The book is well written and convenient to read, and its content is interesting and useful. … it might serve for preparing seminar talks or lectures.” (Wolfgang Bühring, Mathematical Reviews, Issue 2011 j)

“The aim of this textbook is to complete the existing gap in the literature related to the structure of and relationship between different special functions. … In every chapter, many interconnections between the considered special functions are presented. Every chapter is endowed with exercises and problems.” (Boris V. Loginov, Zentralblatt MATH, Vol. 1215, 2011)


Try our Free Online Math Solver!

Differential equations often arise in physics as consequences of Newton's second law:

(recall that the notation means ``take the derivative with respect to t. There is usually some physical description (often resulting in a mathematical model) of the force F -one that relates the force to the position of the moving particle (such as the distance of a massive particle from a gravitational source or of an electrically charged particle from another charged particle) or its velocity (think of air resistance as an example). The other side of the equation is purely mathematical, and if the mass of the particle is constant it involves the derivative of the particle's velocity - in other words, it involves the second derivative of the position of the particle. Thus, the mathematical problem that results from applying Newton's second law often takes the form of a differential equation for the position as a function of time. Since the differential equation involves second derivatives of the position, it is called a second-order differential equation.

If we take the example of a real spring, the motion we see is mimickedby the Java applet below:

Example: Undamped simple harmonic motion.
As a (highly idealized) example, consider the motion of a cart of massM attached to a nearby wall by means of a spring (see the figure). The spring exerts no force when the cart is at its equilibrium positionx = 0. If the cart is displaced by a distance x, then the spring exerts a restoring force proportional to the displacement and opposite in direction, i.e., F = -kx, where k is a positive constant whose magnitude is a measure of the stiffness of the spring (this expression for the force was postulated , and is sometimes called Hooke's law).

By Newton's second law, we have F = Mx´ ´ = -kx.We need a function whose second derivative is a negative multiple of itself. Exponential functions will not do in this situation,since the second derivative of is , and must be positive if r is real.This provokes the idea of considering the exponentials of complex numbers (which we could do), or we could just look for another function.

If you ask Maple to dsolve the differential equation x´ ´ + x = 0, it will tell you that two possible answersare: x = sin t and x = cos t.(Verify this!) The relevant properties of the sine and cosine functions we need are:

Now we can use the stretch rule from the previous section to get that two possible solutions to are and .When you take a course in differential equations, you will learn that the most general solution of

for constants c_1 and c_2.

How are c_1 and c_2 determined?Often they are determined because we know some initial conditions, i.e., where the cart started and how fast it was going (just like in the falling object problems last week). For instance, suppose you know that

From the differential equation, we see that . Fromx(0) = 1, we see that c_2 = 1 (Why?). Then we have to computethe derivative:

Then we can use x´ (0) = 6 to get c_1 = 2. The solution of our initial-value problem is


Welcome to advancedhighermaths.co.uk

A sound understanding of Differential Equations is essential to ensure exam success.

Study at Advanced Higher Maths level will provide excellent preparation for your studies when at university. Some universities may require you to gain a pass at AH Maths to be accepted onto the course of your choice. The AH Maths course is fast paced so please do your very best to keep on top of your studies.

For students looking for extra help with the AH Maths course you may wish to consider subscribing to the fantastic additional exam focused resources available in the Online Study Pack.

To access a wealth of additional free resources by topic please either use the above Search Bar or click HERE selecting on the topic you wish to study.

We hope you find this website useful and wish you the very best of success with your AH Maths course in 2021/22. Please find below:

Advanced Higher Maths Resources

1. About Differential Equations

To learn about Differential Equations please click on any of the Theory Guide links in Section 2 below. For students working from the Maths In Action text book the recommended questions on this topic are given in Section 3. Worksheets including actual SQA Exam Questions are highly recommended.

If you would like more help understanding Differential Equations there are full, easy to follow, step-by-step worked solutions to dozens of AH Maths Past & Practice exam questions on all topics in the AH Maths Online Study Pack. Also included in the Study Pack are full worked solutions to the recommended MIA text book questions. Please give yourself every opportunity for success, speak with your parents, and subscribe to the exam focused Online Study Pack today.

Differential Equations

At AH Maths, four types of Differential Equations are taught.

  1. Variables Separable Differential Equations (In ‘Further Integration’ section)
  2. First Order Linear Differential Equations
  3. Second Order Homogeneous Linear Differential Equations
  4. Second Order Non-Homogeneous Differential Equations

Solving each type above involves a different process.

Variables Separable Differential Equations

Source: SQA AH Maths Paper 2017 Question 9

First Order Linear Differential Equations

Source: SQA AH Maths Paper 2012 Question 15

Second Order Homogeneous Linear Differential Equations

Second Order Non-Homogeneous Differential Equations

Source: SQA AH Maths Paper 2016 Question 15

Worked solutions to the above exam questions are available in the Online Study Pack.

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2. Differential Equations – Exam Worksheet & Theory Guides

Thanks to the SQA and authors for making the excellent AH Maths Worksheet & Theory Guides freely available for all to use. These will prove a fantastic resource in helping consolidate your understanding of AH Maths. Clear, easy to follow, step-by-step worked solutions to all SQA AH Maths Questions in the worksheet below are available in the Online Study Pack.

Worksheet/Theory Guides
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Resource Link
____________________________________________________
Answers
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AH Maths Exam Questions 1Differential Equations (Variables Separable) Exam QuestionsAnswers
AH Maths Exam Questions 2Further Differential Equations Exam QuestionsAnswers
AH Maths Formulae ListAH Maths Fomulae List
Theory Guide 1Differential Equations Theory Guide 1
Theory Guide 2Differential Equations Theory Guide 2
Theory Guide 3 (HSN)Differential Equations Theory Guide 3 (HSN)

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3. Differential Equations – Recommended Text Book Questions

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan text book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic
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Page No
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Exercise
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Recommended Ques
__________________
Notes
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First Order Diff Eqns - General SolnPage 128Exercise 8.1Q1a-jIn the Further Integration Section of the AH Maths Course
First Order Diff Eqns - Particular SolnPage 128Exercise 8.1Q2a-gIn the Further Integration Section of the AH Maths Course
Differential Equations in ContextPage 131Exercise 8.2Q2,4,5,6In the Further Integration Section of the AH Maths Course
1st Order Linear Differential EquationsPage 136Exercise 8.3Q1a,b,2a,3a,b
2nd Order Differential Equations
(Roots Real & Distinct)
Page 140Exercise 8.4Q1a,b,c,2a,b
2nd Order Differential Equations
(Roots Real & Coincident)
Page 141Exercise 8.5Q1a,b,c,2a,b
2nd Order Differential Equations
(Roots Not Real)
Page 142Exercise 8.6Q1a,b,c,2a,b
Non-Homogeneous Differential Equations
(Finding General Solution)
Page 146Exercise 8.9Q1a,b,c
Non-Homogeneous Differential Equations
(Finding Particular Solution)
Page 146Exercise 8.9Q2a,b,c



4. AH Maths Past Paper Exam Worksheets by Topic

Thanks to the SQA for making these available. The worksheets by topic below are an excellent study resource since they are actual SQA past paper exam questions. Clear, easy to follow, step-by-step worked solutions to all SQA AH Maths Questions below are available in the Online Study Pack.

Number
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Topic
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Answers
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1Binomial TheoremAnswers
2Complex NumbersAnswers
3DifferentiationAnswers
4Differentiation (Further)Answers
5Differential Equations - Variables SeparableAnswers
6Differential Equations (Further)Answers
7Functions & GraphsAnswers
8IntegrationAnswers
9Integration (Further)Answers
10MatricesAnswers
11Number Theory - Methods of ProofAnswers
12Number Theory (Further) - Euclidean & Number BasesAnswers
13Partial FractionsAnswers
14Sequences & SeriesAnswers
15 Sequences & Series - Maclaurin Answers
16Systems of EquationsAnswers
17VectorsAnswers

5. AH Maths Past Paper Questions by Topic

Thanks to the SQA for making these available. Questions and answers have been split up by topic for your ease of reference. Clear, easy to follow, step-by-step worked solutions to all SQA AH Maths questions below are available in the Online Study Pack.

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Paper
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.
Marking
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Binomial
Theorem
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Partial
Fractions
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.
Differentiation
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Further Differentiation
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Integration
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Further
Integration
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Functions
& Graphs
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Systems of
Equations
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Complex
Numbers
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Seq &
Series
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Further Seq
& Series
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Matrices
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.
Vectors
__________
Methods
of Proof
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Further No
Theory
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Differential
Equations
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Further
Differential Eqns
_________________
Specimen P1Marking Q2 Q4Q6Q8Q3Q5 Q1 Q7
Specimen P2MarkingQ3Q1 Q2,4,8,10 Q7Q11 Q5Q13Q9 Q6Q12
2019MarkingQ9Q4Q1a,b,6Q1c,5,10Q16bQ16aQ3 Q18Q7,17 Q2Q15Q11,14Q12Q13Q8
2018MarkingQ3Q2Q1bQ1a,c,6,13Q8Q15a Q16aQ4,10Q14Q17Q7,11Q16Q9,12Q5 Q15b
2017MarkingQ1Q2Q3Q11,18Q16Q6Q12Q5Q17Q4,10 Q7Q15Q13Q8Q9Q14
2016MarkingQ3Q13Q1a,bQ1c,11Q13Q9Q12Q4Q8Q2Q6Q7Q14Q5,10 Q16Q15
2015MarkingQ1,9 Q2Q4,6,8Q17Q10Q14 Q13Q3 Q5,11Q15Q12Q7Q18Q16
2014MarkingQ214bQ1,13Q1,4,6Q10,12Q15Q11Q3Q16Q14Q9Q7Q5Q7 Q8
2013MarkingQ1 Q2Q11Q4,6Q8Q13 Q7,10Q17 Q3Q15Q9,12Q5Q16Q14
2012MarkingQ415aQ1Q12,13Q8Q11Q7Q14Q3,16bQ2Q6Q9Q516aQ10 Q15
2011MarkingQ2Q13b,73aQ1,11aQ1,11,16Q6 Q10Q8,13Q5Q4Q15Q12 Q9Q14
2010MarkingQ5 Q1Q13Q15Q3,7Q10 Q16Q2Q9Q4,14Q6Q8,12 Q11
2009MarkingQ8Q14Q1aQ1b,11Q5,7Q9Q1316aQ6Q12Q14Q2Q16Q4Q10Q3Q15
2008MarkingQ8Q4Q10,15Q2,5Q4,9,10Q7Q3 Q16Q1Q12Q6Q14Q11 Q13
2007MarkingQ1Q4Q2Q13Q4,10Q4Q16 Q3,11Q9Q6Q5Q15Q12Q7Q14Q8
MixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixed

6. AH Maths Past & Practice Exam Papers

Thanks to the SQA for making these available. Clear, easy to follow, step-by-step worked solutions to all SQA AH Maths questions below are available in the Online Study Pack.

Year
____
Paper Type
_________________
Exam Paper
______________
Marking Scheme
_______________________________________
2019AH SpecimenSpecimenMarking Scheme
2019Advanced HigherExam PaperMarking Scheme
2018Advanced HigherExam PaperMarking Scheme
2017Advanced HigherExam PaperMarking Scheme
2016Advanced HigherExam PaperMarking Scheme
2016AH SpecimenSpecimenMarking Scheme
2016AH ExemplarExemplarMarking Scheme
2015Advanced HigherExam PaperMarking Scheme
2014Advanced HigherExam PaperMarking Scheme
2013Advanced HigherExam PaperMarking Scheme
2012Advanced HigherExam PaperMarking Scheme
2011Advanced HigherExam PaperMarking Scheme
2010Advanced HigherExam PaperMarking Scheme
2009Advanced HigherExam PaperMarking Scheme
2008Advanced HigherExam PaperMarking Scheme
2007Advanced HigherExam PaperMarking Scheme
2006Advanced HigherExam PaperMarking Scheme
2005Advanced HigherExam PaperMarking Scheme
2004Advanced HigherExam PaperMarking Scheme
2003Advanced HigherExam PaperMarking Scheme
2002Advanced HigherExam PaperMarking Scheme
2001Advanced HigherExam PaperMarking Scheme

7. AH Maths 2020 Specimen Exam Paper

Please find below two Specimen Papers courtesy of the SQA. Clear, easy to follow, step-by-step worked solutions to the SQA AH Maths Specimen Paper available in the Online Study Pack.

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Date
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.
Paper
___________
.
Marking
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Binomial
Theorem
________
Partial
Fractions
________
.
Differentiation
___________
Further Differentiation
___________
.
Integration
___________
Further
Integration
____________
Functions
& Graphs
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Systems of
Equations
____________
Complex
Numbers
__________
Seq &
Series
_________
Further Seq
& Series
____________
.
Matrices
_________
.
Vectors
__________
Methods
of Proof
__________
Further No
Theory
___________
Differential
Equations
____________
Further
Differential Eqns
_________________
June 2019Specimen P1Marking Q2 Q4Q6Q8Q3Q5 Q1 Q7
June 2019Specimen P2MarkingQ3Q1 Q2,4,8,10 Q7Q11 Q5Q13Q9 Q6Q12

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8. AH Maths Prelim & Final Exam Practice Papers

Thanks to the SQA and authors for making these freely available. Please use regularly for revision prior to assessments, tests and the final exam. Clear, easy to follow, step-by-step worked solutions to the first five Practice Papers below are available in the Online Study Pack.

AH Practice Exam Paper
_____________________
Marking
___________
AH Practice Exam Paper
_____________________
Marking
___________
Practice Exam Paper 1HEREPractice Exam Paper 5HERE
Practice Exam Paper 2HEREPractice Exam Paper 6HERE
Practice Exam Paper 3HEREPractice Exam Paper 7HERE
Practice Exam Paper 4HEREPractice Exam Paper 8HERE

9. AH Maths Theory Guides

Thanks to the authors for making the excellent AH Maths Theory Guides freely available for all to use. These will prove a fantastic resource in helping consolidate your understanding of AH Maths.

Topic 1
______________________
Topic 2
___________________
Topic 3
_____________________
Topic 4
___________________
Topic 5
___________________
Topic 6
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Partial Fractions 1Binomial 1Gaussian 1Functions 1Differentiation 1Integration 1
Partial Fractions 2Binomial 2Gaussian 2Functions (HSN)Differentiation 2Integration (HSN)
Partial Fractions (HSN)Binomial (HSN)Gaussian (HSN) Differentiation (HSN)

Topic 1
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Topic 2
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Topic 3
___________________
Topic 4
____________________
Topic 5
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Further Differentiation 1Further Integration 1Complex Numbers 1Sequences & Series 1Methods of Proof
Further Differentiation 2Further Integration 2Complex Numbers 2Sequences & Series 2Proof by Induction
Differentiation (HSN)Integration (HSN)Complex Nos (HSN)Seq & Series (HSN)Methods of Proof (HSN)

Topic 1
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Topic 2
_________________
Topic 3
_____________________
Topic 4
_____________________
Topic 5
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Vectors 1Matrices 1Maclaurin Series 1Differential Eqns 1Further Number Theory
Vectors 2Matrices 2MacLaurin Series 2Differential Eqns 2
Vectors 3Matrices 3Maclaurin Series (HSN)Differential Eqns (HSN)
Vectors (HSN)Matrices (HSN)

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10. AH Maths Course Outline, Formulae Sheets & Check List

Thanks to the SQA and authors for making the excellent resources below freely available. These are fantastic check lists to assess your AH Maths knowledge. Please try to use these regularly for revision prior to tests, prelims and the final exam.

Title
____________________________________
Link
___________
Courtesy
___________________
AH Maths Course Outline & TimingsHERE
SQA AH Maths Exam Formulae ListHERECourtesy of SQA
SQA Higher Maths Exam Formulae ListHERECourtesy of SQA
SQA AH Maths Support NotesHERECourtesy of SQA
AH Maths Complete Check ListHERE

11. Text Book Recommended Timings & Questions – Unit One

Course timings, along with specific text book exercises/questions for Unit One, courtesy of Teejay Publishers can be found HERE .

Partial Fractions

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic
_______________________________
Page Number
_____________
Exercise
_____________
Recommended Questions
_______________________
Comment
________________
Type One - Partial FractionsPage 23Exercise 2.2Q1, 5, 12, 18, 19, 22, 25
Type Two - Partial FractionsPage 24Exercise 2.3Q1, 3, 5, 10, 14, 18
Type Three - Partial FractionsPage 25Exercise 2.4Q1, 5, 7, 9, 11
Algebraic Long Division Worksheet WorksheetWorked Solutions
Partial Fraction - Long DivisionPage 26Exercise 2.5Q1 a, b, e, j, l

Binomial Theorem

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic
____________________________________
Page Number
_____________
Exercise
___________
Recommended Questions
_______________________________
Notes for Lesson
__________________________________________________________________________________
Combinations nCrPage 33Exercise 3.3Q1a,b,c,2a,b,c,4a-d,5a,b,6a,7a,b,d
Expanding - Lesson 1Page 36Exercise 3.4Q1a,b,c,2a,i,ii,iii,iv
Expanding - Lesson 2Page 36Exercise 3.4Q3a-d,4a-fTHEORY - Questions 3 & 4
Finding CoefficientsPage 38Exercise 3.5Q1a,b,c,4a,5a,6
Approximation eg 1.05^5 = ?Page 40Exercise 3.6Q1a,b,c,d
Simplifying General Term (SQA Questions) SQA Questions & AnswersCommon SQA Binomial Questions not in AH Text Book

Systems of Equations

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic
______________________________
Page Number
_____________
Exercise
_______________
Recommended Questions
_______________________
Gaussian EliminationPage 265Exercise 14.4Q1a,b,c,d,2a,b,c
Redundancy & InconsistencyPage 268Exercise 14.6Q1a,b,c,2
Redundancy SQA Question 2016 Q4 (SQA)
Inconsistency SQA Question 2017 Q5 (SQA)
ILL ConditioningPage 274Exercise 14.9Q2a,b,c,d
ILL Conditioning SQA Question 2012 Q14c (SQA)

Functions & Graphs

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic
______________________________
Page Number
_____________
Exercise
___________
Recommended Questions
_______________________
Sketching Modulus Function y = |x|Page 66Exercise 5.2Q1-9
Inverse FunctionsPage 67Exercise 5.3Q1a,c,e,g,i,2a,c,e,3
Odd & Even FunctionsPage 74Exercise 5.8Q3a-l
Vertical Asymptotes & BehaviourPage 75Exercise 5.9Q1a-f
Horizontal & Oblique AsymptotesPage 76Exercise 5.10Q1a,b,f,g,k,l
Sketching GraphsPage 77Exercise 5.11Q1a,c,e,i,k

Differential Calculus

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic
___________________________
Page Number
____________
Exercise
___________
Recommended Questions
_______________________
Derivative from First PrinciplesPage 45Exercise 4.1Q1,3,5,7
The Chain RulePage 48Exercise 4.3Q1a,d,2a,c,3b,4a,5a
The Product RulePage 51Exercise 4.5Q1a-h,Q2b,Q3a-l
The Quotient RulePage 52Exercise 4.6Q1,2,3,4
Differentiation - A Mixture!Page 53Exercise 4.7Q1,2,3,4,5
Sec, Cosec & CotPage 55Exercise 4.8Q1a,b,2a,c,d,3a,c,e,g
Exponential FunctionsPage 58Exercise 4.9Q1a,c,e,2a,3e,4a,b,5a,e
Logarithmic FunctionsPage 58Exercise 4.9Q1k,m,o,q,s,2f,g,3a,b,c,4d,e,5d
Nature & Sketching PolynomialsPage 70Exercise 5.5Q1a,b,c,2a,b
ConcavityPage 73Exercise 5.7Q5a,b,c,Q1a,b
ApplicationsPage 187Ex 11.1Q1a,b,e,f,2a,c,3a,c

Integral Calculus

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic
______________________________________
Page No
__________
Exercise
___________
Recommended Questions
_____________________
Integration (Higher Revision)Page 100Exercise 7.1Q1a-i,2a-i,3a-l,4a-f
Integration by SubstitutionPage 103Exercise 7.2Q1a,c,e,g,i,k,m,o,q,s,u,w
Integration by Substitution - Extra Revision!Page 103Exercise 7.2Q1b,d,f,h,j,l,n,p,r,t,v,x
Further Integration by SubstitutionPage 105Exercise 7.3Q2a,b,c,d,4a,b,c,d
Further Integration by SubstitutionPage 105Exercise 7.3Q6a,b,c,d
Further Int'n by Sub'n - sin^m(x), cos^n(x)Page 105Exercise 7.3Q7a,b,c,d,e,f
Further Integration by Substitution - logsPage 105Exercise 7.3Q11a,b,c,d
Substitution & Definite IntegralsPage 107Exercise 7.4Q1a,c,e,g,i,k
Area between curve & x-axisPage 120Exercise 7.10Q1,3
Area between curve & y-axisPage 120Exercise 7.10Q6,7
Volume - revolved around x-axis SQA Question 2014 Q10 (SQA)
Volume - revolved around y-axis SQA Question 2017 Q16 (SQA)
Volume - revolved around x-axisPage 120Exercise 7.10Q11,12
Applications of Integral CalculusPage 187Exercise 11.1Q4,14

12. Text Book Recommended Timings & Questions – Unit Two

Course timings, along with specific text book exercises/questions for Unit Two, courtesy of Teejay Publishers can be found HERE .

Further Differentiation

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic
_______________________________________
Page Number
_____________
Exercise
______________
Recommended Questions
_____________________
Inverse Trig Functions & Chain RulePage 85Exercise 6.2Q1a,b,c,Q2b,c,dQ3a,d
Inverse Trig Fns & Product/Quotient RulesPage 86Exercise 6.3Q2,Q3
Implicit & Explicit Functions - 1Page 89Exercise 6.4Q1,Q2
Implicit & Explicit Functions - 2Page 89Exercise 6.4Q5,Q9,Q4
Second Derivatives of Implicit FunctionsPage 90Exercise 6.5Q1a,d,f,k(i),6
Logarithmic DifferentiationPage 92Exercise 6.6Q1,Q2
Parametric EquationsPage 95Exercise 6.7Q1a,b,c
Parametric Eqns - Differentiation Page 96Exercise 6.8Q1,2,3
Parametric Eqns - Differentiation (Alternative)Page 96Exercise 6.8Q1(i)
Parametric Eqns - Differentiation (Alternative)Page 96Exercise 6.8Q1(ii),Q2,Q3
Applications of Further DifferentiationPage 193Exercise 11.2Q1,Q2,Q3

Further Integration

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic
______________________________________
Page No
__________
Exercise
_________________
Recommended Questions
__________________________
Integration using Inverse Trig FunctionsPage 111Exercise 7.6Q1,2,3,4a,b
Integration using Partial FractionsPage 113Exercise 7.7Q1a,b,2a,b,3a,b,4a,b,5a,b,6a,b
Integration by Parts - 1Page 116Exercise 7.8Q1a-l
Integration by Parts - 2Page 116Exercise 7.8Q2a,c,d,e,f,g,h
Integration by Parts - 3Page 116Exercise 7.8Q5a,b,Q6a,b
Integration by Parts - Special Cases - 1Page 118Exercise 7.9 Q1a,b,c,d
Integration by Parts - Special Cases - 2Page 118Exercise 7.9Q2a,b,c,d,e
First Order Diff Eqns - General SolnPage 128Exercise 8.1Q1a-j
First Order Diff Eqns - Particular SolnPage 128Exercise 8.1Q2a-g
Differential Equations in ContextPage 131Exercise 8.2Q2,4,5,6

Complex Numbers

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic
_________________________________
Page Number
_____________
Exercise
______________
Recommended Questions
_________________________
Arithmetic with Complex NumbersPage 207Exercise 12.1Q1,2,3,6,7,8
Division & Square Roots of Complex NosPage 209Exercise 12.2Q1a,b,c,2c,e,3a,b,f,5a,b
Argand DiagramsPage 211Exercise 12.3Q3a,b,d,e,f,i,6a,b,f,7a,b,c
Multiplying/Dividing in Polar FormPage 215Exercise 12.5Q1a,b,f,g
De Moivre's TheoremPage 218Exercise 12.6Q1,2,3a,4g,h,i,j
Polynomials & Complex NumbersPage 224Exercise 12.8Q2a,d,3a,b,4,5,6a,b
Loci on the Complex PlanePage 213Exercise 12.4Q1a,b,d,f,j,3a,b,4a,b,c
Expanding Trig FormulaPage 219Exercise 12.6Q5,6,7a
Roots of a Complex NumberPage 222Exercise 12.7Q2a,b,c,d,e,f,1a(i)

Sequences & Series, Sigma Notation

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic
_______________________________
Page Number
_____________
Exercise
______________
Recommended Questions
__________________________
Arithmetic SequencesPage 151Exercise 9.1Q1a-f,2a-f,Q3,Q4,Q6
Finding Sum - Arithmetic SequencePage 153Exercise 9.2Q1a,b,c,Q3a-d,Q4a,b,Q5a
Geometric SequencePage 156Exercise 9.3Q1a-e,Q2,Q3,Q5
Finding Sum - Geometric SequencePage 159Exercise 9.4Q1a-f,Q2a-d,Q3a-d,Q4
Finding Sum to InfinityPage 162Exercise 9.5Q1,2,3,4,6
Sigma NotationPage 168Exercise 10.1Q1a-e,Q2a-e

Number Theory & Proof

Topic
_______________________________
Lessons
__________
Questions
_________
Typed Solutions
_______________
Handwritten Solutions
______________________
Exam Questions - Worked Solutions in Online Study Pack
______________________________________________________
Direct ProofLesson 1Ex 1 & 2 Ex 1 & 2 Handwritten Solns2018-Q9,2015-Q12, 2010-Q8a
Proof by Counterexample Lesson 2Ex 3Ex 3 Typed SolnsEx 3 Handwritten Solns2016-Q10, 2013-Q12, 2008-Q11
Proof by Counterexample Ex 4Ex 4 Typed SolnsEx 4 Handwritten Solns2016-Q10, 2013-Q12, 2008-Q11
Proof by ContradictionLesson 3Ex 5Ex 5 Typed SolnsEx 5 Handwritten Solns2010-Q12
Proof by ContrapositiveLesson 4Ex 6Ex 6 Typed SolnsEx 6 Handwritten Solns2017-Q13
Proof by InductionLesson 5Ex 7Ex 7 Typed SolnsEx 7 Handwritten Solns2014-Q7,2013-Q9,2012-Q16a,2011-Q12,2010-Q8b,2009-Q4,2007-Q12
Proof by Induction - Sigma NotationLesson 6Ex 8Ex 8 Typed SolnsEx 8 Handwritten Solns2018-Q12,2016-Q5, 2013-Q9,2009-Q4

13. Text Book Recommended Timings & Questions – Unit Three

Course timings, along with specific text book exercises/questions for Unit Three, courtesy of Teejay Publishers can be found HERE .

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic
__________________________________
Page Number
_____________
Exercise
______________
Recommended Questions
________________________
Lesson/Notes
_________________
Higher Revision On VectorsPage 282Exercise 15.1Q6,7,8
The Vector Product - 1Page 286Exercise 15.3Q1,2a,b,5,7,8a,b,10Lesson 1
The Vector Product - 2Page 286Exercise 15.3Q3,4,6,12Lesson 2
The Equations of a LinePage 298Exercise 15.8Q1a,b,2a,3a,c,e,5Lesson 3
Vector Equation of a Straight LinePage 298Exercise 15.9Q2Lesson 3
The Equation of a PlanePage 291Exercise 15.5Q1a,b,c,d,2a,b,3,4a,c,9,10Lesson 4
Angle Between 2 PlanesPage 293Exercise 15.6Q1,2,3Lesson 5
Intersection of Line & PlanePage 300Exercise 15.10Q1a,b,c,2a,b,3,4aLesson 6
Intersection of 2 LinesPage 302Exercise 15.11 Q1,2Lesson 7
Intersection of 2 Planes using GaussianPage 303Exercise 15.12Q1,2Lesson 8
Intersection of 2 Planes - AlternativePage 303Exercise 15.12Q1,2
Intersection of 3 PlanesPage 307Exercise 15.3Q1a,c,2a,cLesson 9

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic
__________________________________
Page Number
_____________
Exercise
______________
Recommended Questions
____________________________
Basic Properties & Operations of MatricesPage 231Exercise 13.1Q1,2,3a,4a,c,e,i,p,t,7a,f,9,10
Matrix MultiplicationPage 235Exercise 13.3Q1a,c,2a,c,k,m,o,3a,4,5a,c
Properties of Matrix MultiplicationPage 236Exercise 13.4Q6a,b,7a,b,8a
Determinant of a 2 x 2 MatrixPage 240Exercise 13.6Q1a,b,d,h
Determinant of a 3 x 3 MatrixPage 247Exercise 13.9Q4a,b,c,d,5a,b
Inverse of a 2 x 2 MatrixPage 243Exercise 13.7Q1,2,4,8,9a,b,c
Inverse of a 3 x 3 MatrixPage 275Exercise 14.10Q1a,b,c,d
Transformation MatricesPage 251Exercise 13.10Q1,2,5

Further Sequences & Series (Maclaurin Series)

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic
________________________________
Page Number
_____________
Exercise
______________
Recommended Questions
_______________________
Maclaurin Series for f(x)Page 179Exercise 10.5Q1a,b,c,d,3a,b
Maclaurin Series - Composite FunctionsPage 182Exercise 10.7Q1a,f,2a,3a,6a,7a,8a,b
Maclaurin Series - SQA Questions SQA Questions & Answers

Further Differential Equations

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic
__________________________________
Page Number
_____________
Exercise
______________
Recommended Questions
________________________
1st Order Linear Differential EquationsPage 136Exercise 8.3Q1a,b,2a,3a,b
2nd Order Differential Equations
(Roots Real & Distinct)
Page 140Exercise 8.4Q1a,b,c,2a,b
2nd Order Differential Equations
(Roots Real & Coincident)
Page 141Exercise 8.5Q1a,b,c,2a,b
2nd Order Differential Equations
(Roots Not Real)
Page 142Exercise 8.6Q1a,b,c,2a,b
Non-Homogeneous Differential Equations
(Finding General Solution)
Page 146Exercise 8.9Q1a,b,c
Non-Homogeneous Differential Equations
(Finding Particular Solution)
Page 146Exercise 8.9Q2a,b,c

Further Number Theory & Proof

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic
_______________________________________
Page Number
_____________
Exercise
_________
Recommended Questions
____________________________
Finding the Greatest Common Divisor (GCD)Page 318Ex 16.3Q1a,c,e,g,i
Expressing GCD in the form xa + yb = dPage 320Ex 16.4Q1,2,3,4
Number BasesPage 322Ex 16.5Q1a-d,2a-f
Further Number Theory - SQA Questions SQA Questions & Answers

14. AH Maths Practice Unit Assessments – Solutions Included

Thanks to maths777 for making the excellent resources freely available for all to use. This will prove a fantastic resource in helping you prepare for assessments, tests and the final exam.

Methods in Algebra & Calculus
__________________________
Applications of Algebra & Calculus
____________________________
Geometry, Proof & Systems of Equations
____________________________________
Practice 1Practice 1Practice 1
Practice 2Practice 2Practice 2
Practice 3Practice 3Practice 3

15. AH Maths Video Links

Please click DLB Maths to view AH Maths Past Paper video solutions. There are also many videos showing worked examples by topic on the St Andrews StAnd Maths YouTube Channel link. Both video links are excellent resources in helping you prepare for assessments, tests and the final exam.

16. AH Maths Text Book – Maths In Action (2nd Edition) by Edward Mullan

A fully revised course for the new Curriculum for Excellence examination that is designed to fully support the course’s new structure and unit assessment. A part of the highly regarded Maths in Action series, it provides students with a familiar, clear and carefully structured learning experience that encourages them to build confidence and understanding.

17. Advanced Higher Maths Online Study Pack

Through step-by-step worked solutions to exam questions and recommended MIA text book questions available in the Online Study Pack we cover everything you need to know about Differential Equations to pass your final exam.

For students looking for a ‘good’ pass at AH Maths you may wish to consider subscribing to the fantastic additional exam focused resources available in the Online Study Pack. Subscribing could end up being one of your best ever investments.

Please give yourself every opportunity for success, speak with your parents, and subscribe to the exam focused Online Study Pack today.

We hope the resources on this website prove useful and wish you the very best of success with your AH Maths course in 2022.

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Solve Second Order Differential Equations - part 3

A tutorial on how to solve second order differential equations with auxiliary equation having 2 distinct complex solutions. Examples with detailed solutions are included.

The auxiliary equation of a second order differential equation d 2 y / dx 2 + b dy / dx + c y = 0 is given by
k 2 + b k + c = 0
If b 2 - 4c is < 0, the equation has 2 complex conjugate solution of the form
k1 = r + t i and k2 = r - t i , where i is the imaginary unit.
In such case, it can be shown that the general solution to the second order differential equation may be written as follows
y = e r x [ A cos x t + B sin x t ] where A and B are constants.

Example 1: Solve the second order differential equation given by

Solution to Example 1
The auxiliary equation is given by
k 2 + k + 2 = 0
Solve for k to obtain 2 complex conjugate solutions
k1 = -1 / 2 - i 𕔋 / 2
and k2 = -1 / 2 + i 𕔋 / 2,
r = -1/2 (real part)
and t = 𕔋 /2 (imaginary part)
The general solution to the given differential equation is given by
y = e - x / 2 [ A cos ((𕔋 /2) x) + B sin ((𕔋/2) x) ]
where A and B are constants.

Example 2: Solve the second order differential equation given by

y" + 𕔇 y' + 3 y = 0
with the initial conditions y(0) = 1 and y'(0) = 0

Solution to Example 2
The auxiliary equation is given by
k 2 + 𕔇 k + 3 = 0
Solve the quadratic equation to obtain
k1 = - 𕔇/2 + 3/2 i and k2 = - 𕔇/2 - 3/2 i
The general solution to the given differential equation is given by
y = e -(𕔇/2) x [ A sin (3/2)x + B cos (3/2)x ]
The initial condition y(0) = 1 gives
y(0) = e 0 [ A sin 0 + B cos (0) ] = 1 which gives B = 1
y'(0) = 0 gives
y'(0) = -(𕔇/2)e 0 [ A sin 0 + B cos 0 ] + e 0 [ (3/2) A cos 0 - (3/2) B sin 0 ]
Solve the system of equations B = 1 and -(𕔇/2) B + (3/2) A = 0 to obtain
A = 𕔇/3 and B = 1
The solution may be written as
y = e -(𕔇/2) x [ (𕔇/3) sin (3/2) x + cos (3/2) x]


Second-order ordinary differential equations

Examples of Applications of The Power Series.

Ordinary differential equations of first order

Examples of Differential Equations of Second.

Examples of Systems of Differential Equations.

Stability Theory of Large-Scale Dynamical Systems

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Ordinary differential equations, and second-order equations in particular, are at the heart of many mathematical descriptions of physical systems, as used by engineers, physicists and applied mathematicians. This text provides an introduction to all the relevant material normally encountered at university level: series solution, special functions (Bessel, etc.), Sturm-Liouville theory (involving the appearance of eigenvalues and eigenfunctions) and the definition, properties and use of various integral transforms (Fourier, Laplace, etc.). Numerous worked examples are provided throughout.


19.3 Second Order Differential Equations

A second order differential equation is one that expresses the second derivative of the dependent variable as a function of the variable and its first derivative. (More generally it is an equation involving that variable and its second derivative, and perhaps its first derivative.)

Perhaps the easiest way to handle such an equation is to give a name to the first derivative. Then the original equation becomes a pair of coupled equations for the dependent variable and for its derivative. What you get when doing this is a pair of first order differential equations like the pair of coupled equations seen in the Predator Prey problem.

Given the equation (x'' = f(x,x',t)), we set (z = x') and get the two equations:

Starting with initial values for (y) and (y') we can produce a left hand rule approximate solutions to these equations by keeping track of (y, z) and (z') as (t) increases by some small increment (d). We can plot solutions in three ways, as "orbits" using (x) and (z) as axes, or plot (x) and/or (z) as functions of (t).

The example of forced harmonic motion:

gives rise to the coupled equations

Newton's Laws of motion yield second order differential equations for the positions of objects. There are three dimensions of motion for each particle. They are often reformulated as twice as many first order differential equations, in almost the same way. We will describe this reformulation in one dimension The same thing can be done with any number of dimensions.

In many interesting situations energy is conserved. Energy does not appear in Newton's equation (F = ma). We first have to define it.

The kinetic energy of an object of mass (m) moving in one dimension with speed (v) is (frac<2>). Its momentum, (p), is (mv). (p) rather than (v) is the second variable introduced to reduced the equation to first order.

The kinetic energy is then (frac<2m>). The force (F) on the particle is defined to be the negative of the derivative of the potential energy with respect to the dependent variable (keeping all the other dependent variables and momenta fixed). Thus in the case of gravity on the surface of the earth, the force on an object of weight (m) exerted by the earth is (-mg), and the potential energy is (mgh).

The energy also called the Hamiltonian of the system and written as (H), is the sum of the kinetic and potential energies. (Incidentally, the (H) symbol originally was a Greek capital eta and was chosen to be so because energy begins with E.)

Thus for gravity on the earth’s surface the Hamiltonian is given by.

The equations of motion equivalent to (F=ma) then become:

The quaint symbols (frac) that appear here mean that you take the derivative of (H) with respect to (p) treating the other dependent variable (h) as a constant. This sort of derivative is called the partial derivative of (H) with respect to (p). (In complicated situations, when there are several possible other dependent variables, its meaning depends on which ones you are keeping constant. Here it is well defined.)

Exercise 19.4 What is the Hamiltonian for an undamped and unforced harmonic oscillator (for which the force is (-kx)?


Watch the video: Differentialligninger (October 2021).