# 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

## 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

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:

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 _______________________________ Resource Link ____________________________________________________ Answers ____________ AH Maths Exam Questions 1 Differential Equations (Variables Separable) Exam Questions Answers AH Maths Exam Questions 2 Further Differential Equations Exam Questions Answers AH Maths Formulae List AH Maths Fomulae List Theory Guide 1 Differential Equations Theory Guide 1 Theory Guide 2 Differential 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 ______________________________________ Page No __________ Exercise ____________ Recommended Ques __________________ Notes _____________________________________________________ First Order Diff Eqns - General Soln Page 128 Exercise 8.1 Q1a-j In the Further Integration Section of the AH Maths Course First Order Diff Eqns - Particular Soln Page 128 Exercise 8.1 Q2a-g In the Further Integration Section of the AH Maths Course Differential Equations in Context Page 131 Exercise 8.2 Q2,4,5,6 In the Further Integration Section of the AH Maths Course 1st Order Linear Differential Equations Page 136 Exercise 8.3 Q1a,b,2a,3a,b 2nd Order Differential Equations (Roots Real & Distinct) Page 140 Exercise 8.4 Q1a,b,c,2a,b 2nd Order Differential Equations (Roots Real & Coincident) Page 141 Exercise 8.5 Q1a,b,c,2a,b 2nd Order Differential Equations (Roots Not Real) Page 142 Exercise 8.6 Q1a,b,c,2a,b Non-Homogeneous Differential Equations (Finding General Solution) Page 146 Exercise 8.9 Q1a,b,c Non-Homogeneous Differential Equations (Finding Particular Solution) Page 146 Exercise 8.9 Q2a,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.

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.

 . Paper ___________ . Marking ______ Binomial Theorem ________ Partial Fractions ________ . Differentiation ___________ Further Differentiation ___________ . Integration ___________ Further Integration ____________ Functions & Graphs ___________ Systems of Equations ____________ Complex Numbers __________ Seq & Series _________ Further Seq & Series ____________ . Matrices _________ . Vectors __________ Methods of Proof __________ Further No Theory ___________ Differential Equations ____________ Further Differential Eqns _________________ Specimen P1 Marking Q2 Q4 Q6 Q8 Q3 Q5 Q1 Q7 Specimen P2 Marking Q3 Q1 Q2,4,8,10 Q7 Q11 Q5 Q13 Q9 Q6 Q12 2019 Marking Q9 Q4 Q1a,b,6 Q1c,5,10 Q16b Q16a Q3 Q18 Q7,17 Q2 Q15 Q11,14 Q12 Q13 Q8 2018 Marking Q3 Q2 Q1b Q1a,c,6,13 Q8 Q15a Q16a Q4,10 Q14 Q17 Q7,11 Q16 Q9,12 Q5 Q15b 2017 Marking Q1 Q2 Q3 Q11,18 Q16 Q6 Q12 Q5 Q17 Q4,10 Q7 Q15 Q13 Q8 Q9 Q14 2016 Marking Q3 Q13 Q1a,b Q1c,11 Q13 Q9 Q12 Q4 Q8 Q2 Q6 Q7 Q14 Q5,10 Q16 Q15 2015 Marking Q1,9 Q2 Q4,6,8 Q17 Q10 Q14 Q13 Q3 Q5,11 Q15 Q12 Q7 Q18 Q16 2014 Marking Q2 14b Q1,13 Q1,4,6 Q10,12 Q15 Q11 Q3 Q16 Q14 Q9 Q7 Q5 Q7 Q8 2013 Marking Q1 Q2 Q11 Q4,6 Q8 Q13 Q7,10 Q17 Q3 Q15 Q9,12 Q5 Q16 Q14 2012 Marking Q4 15a Q1 Q12,13 Q8 Q11 Q7 Q14 Q3,16b Q2 Q6 Q9 Q5 16a Q10 Q15 2011 Marking Q2 Q1 3b,7 3a Q1,11a Q1,11,16 Q6 Q10 Q8,13 Q5 Q4 Q15 Q12 Q9 Q14 2010 Marking Q5 Q1 Q13 Q15 Q3,7 Q10 Q16 Q2 Q9 Q4,14 Q6 Q8,12 Q11 2009 Marking Q8 Q14 Q1a Q1b,11 Q5,7 Q9 Q13 16a Q6 Q12 Q14 Q2 Q16 Q4 Q10 Q3 Q15 2008 Marking Q8 Q4 Q10,15 Q2,5 Q4,9,10 Q7 Q3 Q16 Q1 Q12 Q6 Q14 Q11 Q13 2007 Marking Q1 Q4 Q2 Q13 Q4,10 Q4 Q16 Q3,11 Q9 Q6 Q5 Q15 Q12 Q7 Q14 Q8 Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed

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.

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.

 . Date __________ . Paper ___________ . Marking ______ Binomial Theorem ________ Partial Fractions ________ . Differentiation ___________ Further Differentiation ___________ . Integration ___________ Further Integration ____________ Functions & Graphs ___________ Systems of Equations ____________ Complex Numbers __________ Seq & Series _________ Further Seq & Series ____________ . Matrices _________ . Vectors __________ Methods of Proof __________ Further No Theory ___________ Differential Equations ____________ Further Differential Eqns _________________ June 2019 Specimen P1 Marking Q2 Q4 Q6 Q8 Q3 Q5 Q1 Q7 June 2019 Specimen P2 Marking Q3 Q1 Q2,4,8,10 Q7 Q11 Q5 Q13 Q9 Q6 Q12

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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 1 HERE Practice Exam Paper 5 HERE Practice Exam Paper 2 HERE Practice Exam Paper 6 HERE Practice Exam Paper 3 HERE Practice Exam Paper 7 HERE Practice Exam Paper 4 HERE Practice Exam Paper 8 HERE

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 ___________________ Partial Fractions 1 Binomial 1 Gaussian 1 Functions 1 Differentiation 1 Integration 1 Partial Fractions 2 Binomial 2 Gaussian 2 Functions (HSN) Differentiation 2 Integration (HSN) Partial Fractions (HSN) Binomial (HSN) Gaussian (HSN) Differentiation (HSN)

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

 Topic 1 ________________________ Topic 2 _________________ Topic 3 _____________________ Topic 4 _____________________ Topic 5 ______________________________ Vectors 1 Matrices 1 Maclaurin Series 1 Differential Eqns 1 Further Number Theory Vectors 2 Matrices 2 MacLaurin Series 2 Differential Eqns 2 Vectors 3 Matrices 3 Maclaurin 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 & Timings HERE SQA AH Maths Exam Formulae List HERE Courtesy of SQA SQA Higher Maths Exam Formulae List HERE Courtesy of SQA SQA AH Maths Support Notes HERE Courtesy of SQA AH Maths Complete Check List HERE

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 Fractions Page 23 Exercise 2.2 Q1, 5, 12, 18, 19, 22, 25 Type Two - Partial Fractions Page 24 Exercise 2.3 Q1, 3, 5, 10, 14, 18 Type Three - Partial Fractions Page 25 Exercise 2.4 Q1, 5, 7, 9, 11 Algebraic Long Division Worksheet Worksheet Worked Solutions Partial Fraction - Long Division Page 26 Exercise 2.5 Q1 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 nCr Page 33 Exercise 3.3 Q1a,b,c,2a,b,c,4a-d,5a,b,6a,7a,b,d Expanding - Lesson 1 Page 36 Exercise 3.4 Q1a,b,c,2a,i,ii,iii,iv Expanding - Lesson 2 Page 36 Exercise 3.4 Q3a-d,4a-f THEORY - Questions 3 & 4 Finding Coefficients Page 38 Exercise 3.5 Q1a,b,c,4a,5a,6 Approximation eg 1.05^5 = ? Page 40 Exercise 3.6 Q1a,b,c,d Simplifying General Term (SQA Questions) SQA Questions & Answers Common 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 Elimination Page 265 Exercise 14.4 Q1a,b,c,d,2a,b,c Redundancy & Inconsistency Page 268 Exercise 14.6 Q1a,b,c,2 Redundancy SQA Question 2016 Q4 (SQA) Inconsistency SQA Question 2017 Q5 (SQA) ILL Conditioning Page 274 Exercise 14.9 Q2a,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 66 Exercise 5.2 Q1-9 Inverse Functions Page 67 Exercise 5.3 Q1a,c,e,g,i,2a,c,e,3 Odd & Even Functions Page 74 Exercise 5.8 Q3a-l Vertical Asymptotes & Behaviour Page 75 Exercise 5.9 Q1a-f Horizontal & Oblique Asymptotes Page 76 Exercise 5.10 Q1a,b,f,g,k,l Sketching Graphs Page 77 Exercise 5.11 Q1a,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 Principles Page 45 Exercise 4.1 Q1,3,5,7 The Chain Rule Page 48 Exercise 4.3 Q1a,d,2a,c,3b,4a,5a The Product Rule Page 51 Exercise 4.5 Q1a-h,Q2b,Q3a-l The Quotient Rule Page 52 Exercise 4.6 Q1,2,3,4 Differentiation - A Mixture! Page 53 Exercise 4.7 Q1,2,3,4,5 Sec, Cosec & Cot Page 55 Exercise 4.8 Q1a,b,2a,c,d,3a,c,e,g Exponential Functions Page 58 Exercise 4.9 Q1a,c,e,2a,3e,4a,b,5a,e Logarithmic Functions Page 58 Exercise 4.9 Q1k,m,o,q,s,2f,g,3a,b,c,4d,e,5d Nature & Sketching Polynomials Page 70 Exercise 5.5 Q1a,b,c,2a,b Concavity Page 73 Exercise 5.7 Q5a,b,c,Q1a,b Applications Page 187 Ex 11.1 Q1a,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 100 Exercise 7.1 Q1a-i,2a-i,3a-l,4a-f Integration by Substitution Page 103 Exercise 7.2 Q1a,c,e,g,i,k,m,o,q,s,u,w Integration by Substitution - Extra Revision! Page 103 Exercise 7.2 Q1b,d,f,h,j,l,n,p,r,t,v,x Further Integration by Substitution Page 105 Exercise 7.3 Q2a,b,c,d,4a,b,c,d Further Integration by Substitution Page 105 Exercise 7.3 Q6a,b,c,d Further Int'n by Sub'n - sin^m(x), cos^n(x) Page 105 Exercise 7.3 Q7a,b,c,d,e,f Further Integration by Substitution - logs Page 105 Exercise 7.3 Q11a,b,c,d Substitution & Definite Integrals Page 107 Exercise 7.4 Q1a,c,e,g,i,k Area between curve & x-axis Page 120 Exercise 7.10 Q1,3 Area between curve & y-axis Page 120 Exercise 7.10 Q6,7 Volume - revolved around x-axis SQA Question 2014 Q10 (SQA) Volume - revolved around y-axis SQA Question 2017 Q16 (SQA) Volume - revolved around x-axis Page 120 Exercise 7.10 Q11,12 Applications of Integral Calculus Page 187 Exercise 11.1 Q4,14

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 Rule Page 85 Exercise 6.2 Q1a,b,c,Q2b,c,dQ3a,d Inverse Trig Fns & Product/Quotient Rules Page 86 Exercise 6.3 Q2,Q3 Implicit & Explicit Functions - 1 Page 89 Exercise 6.4 Q1,Q2 Implicit & Explicit Functions - 2 Page 89 Exercise 6.4 Q5,Q9,Q4 Second Derivatives of Implicit Functions Page 90 Exercise 6.5 Q1a,d,f,k(i),6 Logarithmic Differentiation Page 92 Exercise 6.6 Q1,Q2 Parametric Equations Page 95 Exercise 6.7 Q1a,b,c Parametric Eqns - Differentiation Page 96 Exercise 6.8 Q1,2,3 Parametric Eqns - Differentiation (Alternative) Page 96 Exercise 6.8 Q1(i) Parametric Eqns - Differentiation (Alternative) Page 96 Exercise 6.8 Q1(ii),Q2,Q3 Applications of Further Differentiation Page 193 Exercise 11.2 Q1,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 Functions Page 111 Exercise 7.6 Q1,2,3,4a,b Integration using Partial Fractions Page 113 Exercise 7.7 Q1a,b,2a,b,3a,b,4a,b,5a,b,6a,b Integration by Parts - 1 Page 116 Exercise 7.8 Q1a-l Integration by Parts - 2 Page 116 Exercise 7.8 Q2a,c,d,e,f,g,h Integration by Parts - 3 Page 116 Exercise 7.8 Q5a,b,Q6a,b Integration by Parts - Special Cases - 1 Page 118 Exercise 7.9 Q1a,b,c,d Integration by Parts - Special Cases - 2 Page 118 Exercise 7.9 Q2a,b,c,d,e First Order Diff Eqns - General Soln Page 128 Exercise 8.1 Q1a-j First Order Diff Eqns - Particular Soln Page 128 Exercise 8.1 Q2a-g Differential Equations in Context Page 131 Exercise 8.2 Q2,4,5,6

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 Numbers Page 207 Exercise 12.1 Q1,2,3,6,7,8 Division & Square Roots of Complex Nos Page 209 Exercise 12.2 Q1a,b,c,2c,e,3a,b,f,5a,b Argand Diagrams Page 211 Exercise 12.3 Q3a,b,d,e,f,i,6a,b,f,7a,b,c Multiplying/Dividing in Polar Form Page 215 Exercise 12.5 Q1a,b,f,g De Moivre's Theorem Page 218 Exercise 12.6 Q1,2,3a,4g,h,i,j Polynomials & Complex Numbers Page 224 Exercise 12.8 Q2a,d,3a,b,4,5,6a,b Loci on the Complex Plane Page 213 Exercise 12.4 Q1a,b,d,f,j,3a,b,4a,b,c Expanding Trig Formula Page 219 Exercise 12.6 Q5,6,7a Roots of a Complex Number Page 222 Exercise 12.7 Q2a,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 Sequences Page 151 Exercise 9.1 Q1a-f,2a-f,Q3,Q4,Q6 Finding Sum - Arithmetic Sequence Page 153 Exercise 9.2 Q1a,b,c,Q3a-d,Q4a,b,Q5a Geometric Sequence Page 156 Exercise 9.3 Q1a-e,Q2,Q3,Q5 Finding Sum - Geometric Sequence Page 159 Exercise 9.4 Q1a-f,Q2a-d,Q3a-d,Q4 Finding Sum to Infinity Page 162 Exercise 9.5 Q1,2,3,4,6 Sigma Notation Page 168 Exercise 10.1 Q1a-e,Q2a-e

Number Theory & Proof

 Topic _______________________________ Lessons __________ Questions _________ Typed Solutions _______________ Handwritten Solutions ______________________ Exam Questions - Worked Solutions in Online Study Pack ______________________________________________________ Direct Proof Lesson 1 Ex 1 & 2 Ex 1 & 2 Handwritten Solns 2018-Q9,2015-Q12, 2010-Q8a Proof by Counterexample Lesson 2 Ex 3 Ex 3 Typed Solns Ex 3 Handwritten Solns 2016-Q10, 2013-Q12, 2008-Q11 Proof by Counterexample Ex 4 Ex 4 Typed Solns Ex 4 Handwritten Solns 2016-Q10, 2013-Q12, 2008-Q11 Proof by Contradiction Lesson 3 Ex 5 Ex 5 Typed Solns Ex 5 Handwritten Solns 2010-Q12 Proof by Contrapositive Lesson 4 Ex 6 Ex 6 Typed Solns Ex 6 Handwritten Solns 2017-Q13 Proof by Induction Lesson 5 Ex 7 Ex 7 Typed Solns Ex 7 Handwritten Solns 2014-Q7,2013-Q9,2012-Q16a,2011-Q12,2010-Q8b,2009-Q4,2007-Q12 Proof by Induction - Sigma Notation Lesson 6 Ex 8 Ex 8 Typed Solns Ex 8 Handwritten Solns 2018-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 Vectors Page 282 Exercise 15.1 Q6,7,8 The Vector Product - 1 Page 286 Exercise 15.3 Q1,2a,b,5,7,8a,b,10 Lesson 1 The Vector Product - 2 Page 286 Exercise 15.3 Q3,4,6,12 Lesson 2 The Equations of a Line Page 298 Exercise 15.8 Q1a,b,2a,3a,c,e,5 Lesson 3 Vector Equation of a Straight Line Page 298 Exercise 15.9 Q2 Lesson 3 The Equation of a Plane Page 291 Exercise 15.5 Q1a,b,c,d,2a,b,3,4a,c,9,10 Lesson 4 Angle Between 2 Planes Page 293 Exercise 15.6 Q1,2,3 Lesson 5 Intersection of Line & Plane Page 300 Exercise 15.10 Q1a,b,c,2a,b,3,4a Lesson 6 Intersection of 2 Lines Page 302 Exercise 15.11 Q1,2 Lesson 7 Intersection of 2 Planes using Gaussian Page 303 Exercise 15.12 Q1,2 Lesson 8 Intersection of 2 Planes - Alternative Page 303 Exercise 15.12 Q1,2 Intersection of 3 Planes Page 307 Exercise 15.3 Q1a,c,2a,c Lesson 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 Matrices Page 231 Exercise 13.1 Q1,2,3a,4a,c,e,i,p,t,7a,f,9,10 Matrix Multiplication Page 235 Exercise 13.3 Q1a,c,2a,c,k,m,o,3a,4,5a,c Properties of Matrix Multiplication Page 236 Exercise 13.4 Q6a,b,7a,b,8a Determinant of a 2 x 2 Matrix Page 240 Exercise 13.6 Q1a,b,d,h Determinant of a 3 x 3 Matrix Page 247 Exercise 13.9 Q4a,b,c,d,5a,b Inverse of a 2 x 2 Matrix Page 243 Exercise 13.7 Q1,2,4,8,9a,b,c Inverse of a 3 x 3 Matrix Page 275 Exercise 14.10 Q1a,b,c,d Transformation Matrices Page 251 Exercise 13.10 Q1,2,5

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 179 Exercise 10.5 Q1a,b,c,d,3a,b Maclaurin Series - Composite Functions Page 182 Exercise 10.7 Q1a,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 Equations Page 136 Exercise 8.3 Q1a,b,2a,3a,b 2nd Order Differential Equations (Roots Real & Distinct) Page 140 Exercise 8.4 Q1a,b,c,2a,b 2nd Order Differential Equations (Roots Real & Coincident) Page 141 Exercise 8.5 Q1a,b,c,2a,b 2nd Order Differential Equations (Roots Not Real) Page 142 Exercise 8.6 Q1a,b,c,2a,b Non-Homogeneous Differential Equations (Finding General Solution) Page 146 Exercise 8.9 Q1a,b,c Non-Homogeneous Differential Equations (Finding Particular Solution) Page 146 Exercise 8.9 Q2a,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 318 Ex 16.3 Q1a,c,e,g,i Expressing GCD in the form xa + yb = d Page 320 Ex 16.4 Q1,2,3,4 Number Bases Page 322 Ex 16.5 Q1a-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 1 Practice 1 Practice 1 Practice 2 Practice 2 Practice 2 Practice 3 Practice 3 Practice 3

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.

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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

• Bite-sized format (1-2hr reading time)
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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)?