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Mathematical Methods for Physics and Engineering: A Comprehensive Guide

K. F. Riley, M. P. Hobson, S. J. Bence

Chapter 15

Higher-order ordinary differential equations - all with Video Answers

Educators

Chapter Questions

02:28

Problem 1

A simple harmonic oscillator, with natural frequency $\omega_{0}$, experiences an oscillating driving force $f(t)=\cos \omega t .$ Therefore, its equation of motion is
$$
\frac{d^{2} x}{d t^{2}}+\omega_{0}^{2} x=\cos \omega t
$$
where $x$ is its position. Given that at $t=0$ we have $x=d x / d t=0$, find the function $x(t) .$ Describe the solution if $\omega$ is approximately, but not exactly, equal to $\omega_{0}$.

Adriano Chikande
Adriano Chikande
Numerade Educator
08:10

Problem 2

Find the roots of the auxiliary equation for the following. Hence solve them for the boundary conditions stated.
(a) $\frac{d^{2} f}{d t^{2}}+2 \frac{d f}{d t}+5 f=0 \quad$ with $f(0)=1, f^{\prime}(0)=0$.
(b) $\frac{d^{2} f}{d t^{2}}+2 \frac{d f}{d t}+5 f=e^{-t} \cos 3 t \quad$ with $f(0)=0, f^{\prime}(0)=0$.

Keshav Singh
Keshav Singh
Numerade Educator
04:56

Problem 3

The theory of bent beams shows that at any point in the beam the 'bending moment' is given by $K / \rho$, where $K$ is a constant (that depends upon the beam material and cross-sectional shape) and $\rho$ is the radius of curvature at that point. Consider a light beam of length $L$ whose ends, $x=0$ and $x=L$, are supported at the same vertical height and which has a weight $W$ suspended from its centre. Verify that at any point $x(0 \leq x \leq L / 2$ for definiteness) the net magnitude of the bending moments, (bending moment $=$ force $\times$ perpendicular distance) due to the weight and support reactions, evaluated on either side of $x$, is $W x / 2$.
If the beam is only slightly bent, so that $(d y / d x)^{2} \ll 1$, where $y=y(x)$ is the downward displacement of the beam at $x$, show that the beam profile satisfies the approximate equation
$$
\frac{d^{2} y}{d x^{2}}=-\frac{W x}{2 K}
$$
By integrating this equation twice and using physically imposed conditions on your solution at $x=0$ and $x=L / 2$, show that the downward displacement at the centre of the beam is $W L^{3} /(48 K)$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:58

Problem 4

Solve the differential equation
$$
\frac{d^{2} f}{d t^{2}}+6 \frac{d f}{d t}+9 f=e^{-t}
$$
subject to the conditions $f=0$ and $d f / d t=\lambda$ at $t=0$
Find the equation satisfied by the positions of the turning points of $f(t)$ and hence, by drawing suitable sketch graphs, determine the number of turning points the solution has in the range $t>0$ if (a) $\lambda=1 / 4$, and (b) $\lambda=-1 / 4$.

Dwijendra Rao
Dwijendra Rao
Numerade Educator
View

Problem 5

The function $f(t)$ satisfies the differential equation
$$
\frac{d^{2} f}{d t^{2}}+8 \frac{d f}{d t}+12 f=12 e^{-4 t}
$$
For the following sets of boundary conditions determine whether it has solutions, and, if so, find them:
(a) $f(0)=0, \quad f^{\prime}(0)=0, \quad f(\ln \sqrt{2})=0$
(b) $f(0)=0, \quad f^{\prime}(0)=-2, \quad f(\ln \sqrt{2})=0 .$

Nick Johnson
Nick Johnson
Numerade Educator
05:43

Problem 6

Determine the values of $\alpha$ and $\beta$ for which the following functions are linearly dependent:
$$
\begin{aligned}
&y_{1}(x)=x \cosh x+\sinh x \\
&y_{2}(x)=x \sinh x+\cosh x \\
&y_{3}(x)=(x+\alpha) e^{x} \\
&y_{4}(x)=(x+\beta) e^{-x}
\end{aligned}
$$
You will find it convenient to work with those linear combinations of the $y_{i}(x)$ that can be written the most compactly.

Keshav Singh
Keshav Singh
Numerade Educator
01:03

Problem 7

A solution of the differential equation
$$
\frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}+y=4 e^{-x}
$$
takes the value 1 when $x=0$ and the value $e^{-1}$ when $x=1$. What is its value when $x=2 ?$

Tyler Moulton
Tyler Moulton
Numerade Educator
02:54

Problem 8

The two functions $x(t)$ and $y(t)$ satisfy the simultaneous equations
$$
\begin{aligned}
&\frac{d x}{d t}-2 y=-\sin t \\
&\frac{d y}{d t}+2 x=5 \cos t
\end{aligned}
$$
Find explicit expressions for $x(t)$ and $y(t)$, given that $x(0)=3$ and $y(0)=2$. Sketch the solution trajectory in the $x y$-plane for $0 \leq t<2 \pi$, showing that the trajectory crosses itself at $(0,1 / 2)$ and passes through the points $(0,-3)$ and $(0,-1)$ in the negative $x$-direction.

Bobby Barnes
Bobby Barnes
University of North Texas
01:05

Problem 9

Find the general solutions of
(a) $\frac{d^{3} y}{d x^{3}}-12 \frac{d y}{d x}+16 y=32 x-8$,
(b) $\frac{d}{d x}\left(\frac{1}{y} \frac{d y}{d x}\right)+(2 a \operatorname{coth} 2 a x)\left(\frac{1}{y} \frac{d y}{d x}\right)=2 a^{2}$,
where $a$ is a constant.

Tyler Moulton
Tyler Moulton
Numerade Educator
04:10

Problem 10

Use the method of Laplace transforms to solve
(a) $\frac{d^{2} f}{d t^{2}}+5 \frac{d f}{d t}+6 f=0, \quad f(0)=1, f^{\prime}(0)=-4$,
(b) $\frac{d^{2} f}{d t^{2}}+2 \frac{d f}{d t}+5 f=0, \quad f(0)=1, f^{\prime}(0)=0$.

Keshav Singh
Keshav Singh
Numerade Educator
02:06

Problem 11

The quantities $x(t), y(t)$ satisfy the simultaneous equations
$$
\begin{aligned}
&\ddot{x}+2 n \dot{x}+n^{2} x=0 \\
&\ddot{y}+2 n \dot{y}+n^{2} y=\mu \dot{x}
\end{aligned}
$$
where $x(0)=y(0)=\dot{y}(0)=0$ and $\dot{x}(0)=\lambda$. Show that
$$
y(t)=\frac{1}{2} \mu \lambda t^{2}\left(1-\frac{1}{3} n t\right) \exp (-n t).
$$

Narayan Hari
Narayan Hari
Numerade Educator
05:03

Problem 12

Use Laplace transforms to solve, for $t \geq 0$, the differential equations
$$
\begin{aligned}
\ddot{x}+2 x+y &=\cos t \\
\ddot{y}+2 x+3 y &=2 \cos t
\end{aligned}
$$
which describe a coupled system that starts from rest at the equilibrium position. Show that the subsequent motion takes place along a straight line in the $x y$-plane. Verify that the frequency at which the system is driven is equal to one of the resonance frequencies of the system; explain why there is no resonant behaviour in the solution you have obtained

Keshav Singh
Keshav Singh
Numerade Educator
08:10

Problem 13

Two unstable isotopes $A$ and $B$ and a stable isotope $C$ have the following decay rates per atom present: $A \rightarrow B, 3 \mathrm{~s}^{-1} ; A \rightarrow C, 1 \mathrm{~s}^{-1} ; B \rightarrow C, 2 \mathrm{~s}^{-1}$. Initially a quantity $x_{0}$ of $A$ is present and none of the other two types. Using Laplace transforms, find the amount of $C$ present at a later time $t$.

Logan Heckart
Logan Heckart
Numerade Educator
02:17

Problem 14

For a lightly damped $\left(\gamma<\omega_{0}\right)$ harmonic oscillator driven at its undamped resonance frequency $\omega_{0}$, the displacement $x(t)$ at time $t$ satisfies the equation
$$
\frac{d^{2} x}{d t^{2}}+2 \gamma \frac{d x}{d t}+\omega_{0}^{2} x=F \sin \omega_{0} t.
$$
Use Laplace transforms to find the displacement at a general time if the oscillator starts from rest at its equilibrium position.
(a) Show that ultimately the oscillation has amplitude $F /\left(2 \omega_{0} \gamma\right)$ with a phase lag of $\pi / 2$ relative to the driving force $F$.
(b) By differentiating the original equation, conclude that if $x(t)$ is expanded as a power series in $t$ for small $t$ then the first non-vanishing term is $F \omega_{0} t^{3} / 6$. Confirm this conclusion by expanding your explicit solution.

Surendra Kumar
Surendra Kumar
Numerade Educator
02:02

Problem 15

The 'golden mean', which is said to describe the most aesthetically pleasing proportions for the sides of a rectangle (e.g. the ideal picture frame), is given by the limiting value of the ratio of successive terms of the Fibonacci series $u_{n}$, which is generated by
$$
u_{n+2}=u_{n+1}+u_{n}
$$
with $u_{0}=0$ and $u_{1}=1$. Find an expression for the general term of the series and verify that the golden mean is equal to the larger root of the recurrence relation's characteristic equation.

James Kiss
James Kiss
Numerade Educator
04:33

Problem 16

In a particular scheme for modelling numerically one-dimensional fluid flow, the successive values, $u_{n}$, of the solution are connected for $n \geq 1$ by the difference equation
$$
c\left(u_{n+1}-u_{n-1}\right)=d\left(u_{n+1}-2 u_{n}+u_{n-1}\right)
$$
where $c$ and $d$ are positive constants. The boundary conditions are $u_{0}=0$ and $u_{M}=1$. Find the solution to the equation and show that successive values of $u_{n}$ will have alternating signs if $c>d$.

Keshav Singh
Keshav Singh
Numerade Educator
09:01

Problem 17

The first few terms of a series $u_{n}$, starting with $u_{0}$, are $1,2,2,1,6,-3$. The series is generated by a recurrence relation of the form
$$
u_{n}=P u_{n-2}+Q u_{n-4}
$$
where $P$ and $Q$ are constants. Find an expression for the general term of the series and show that the series in fact consists of two other interleaved series given by
$$
\begin{array}{r}
u_{2 m}=\frac{2}{3}+\frac{1}{3} 4^{m} \\
u_{2 m+1}=\frac{7}{3}-\frac{1}{3} 4^{m}
\end{array}
$$
for $m=0,1,2, \ldots$

Sandip Ranjan
Sandip Ranjan
Numerade Educator
03:53

Problem 18

Find an explicit expression for the $u_{n}$ satisfying
$$
u_{n+1}+5 u_{n}+6 u_{n-1}=2^{n}
$$
given that $u_{0}=u_{1}=1$. Deduce that $2^{n}-26(-3)^{n}$ is divisible by 5 for all integer $n$.

Keshav Singh
Keshav Singh
Numerade Educator
01:44

Problem 19

Find the general expression for the $u_{n}$ satisfying
$$
u_{n+1}=2 u_{n-2}-u_{n-1}
$$
with $u_{0}=u_{1}=0$ and $u_{2}=1$, and show that they can be written in the form
$$
u_{n}=\frac{1}{5}-\frac{2^{n / 2}}{\sqrt{5}} \cos \left(\frac{3 \pi n}{4}-\phi\right)
$$
where $\tan \phi=2$.

Heather Zimmers
Heather Zimmers
Numerade Educator
12:19

Problem 20

Consider the seventh-order recurrence relation
$$
u_{n+7}-u_{n+6}-u_{n+5}+u_{n+4}-u_{n+3}+u_{n+2}+u_{n+1}-u_{n}=0
$$
Find the most general form of its solution, and show that:
(a) if only the four initial values $u_{0}=0, u_{1}=2, u_{3}=6$ and $u_{3}=12$, are specified, the relation has one solution which cycles repeatedly through this set of four numbers.
(b) but if, in addition, it is required that $u_{4}=20, u_{5}=30$ and $u_{6}=42$ then the solution is unique, with $u_{n}=n(n+1)$.

Bryan Lynn
Bryan Lynn
Numerade Educator
04:09

Problem 21

Find the general solution of
$$
x^{2} \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}+y=x
$$
given that $y(1)=1$ and $y(e)=2 e$.

Sam Stansfield
Sam Stansfield
Numerade Educator
05:06

Problem 22

Find the general solution of
$$
(x+1)^{2} \frac{d^{2} y}{d x^{2}}+3(x+1) \frac{d y}{d x}+y=x^{2}.
$$

Keshav Singh
Keshav Singh
Numerade Educator
01:53

Problem 23

Prove that the general solution of
$$
(x-2) \frac{d^{2} y}{d x^{2}}+3 \frac{d y}{d x}+\frac{4 y}{x^{2}}=0
$$
is given by
$$
y(x)=\frac{1}{(x-2)^{2}}\left[k\left(\frac{2}{3 x}-\frac{1}{2}\right)+c x^{2}\right].
$$

Madeline Zackeo
Madeline Zackeo
Numerade Educator
08:59

Problem 24

Use the method of variation of parameters to find the general solutions of
(a) $\frac{d^{2} y}{d x^{2}}-y=x^{n}$,
(b) $\frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+y=2 x e^{x}$.

Keshav Singh
Keshav Singh
Numerade Educator
01:01

Problem 25

Use the intermediate result of exercise $15.24(\mathrm{a})$ to find the Green's function which satisfies
$$
\frac{d^{2} G(x, \xi)}{d x^{2}}-G(x, \xi)=\delta(x-\xi) \quad \text { with } \quad G(0, \xi)=G(1, \xi)=0.
$$

Carson Merrill
Carson Merrill
Numerade Educator
07:54

Problem 26

(a) Given that $y_{1}(x)=1 / x$ is a solution of
$$
F(x, y)=x(x+1) \frac{d^{2} y}{d x^{2}}+\left(2-x^{2}\right) \frac{d y}{d x}-(2+x) y=0
$$
find a second linearly independent solution,
(i) by setting $y_{2}(x)=y_{1}(x) u(x)$,
(ii) by noting the sum of the coefficients in the equation.
(b) Hence, using the variation of parameters method, find the general solution of
$$
F(x, y)=(x+1)^{2}.
$$

Keshav Singh
Keshav Singh
Numerade Educator
03:26

Problem 27

Show generally that if $y_{1}(x)$ and $y_{2}(x)$ are linearly independent solutions of
$$
\frac{d^{2} y}{d x^{2}}+p(x) \frac{d y}{d x}+q(x) y=0
$$
with $y_{1}(0)=0$ and $y_{2}(1)=0$, then the Green's function $G(x, \xi)$ for the interval $0 \leq x, \xi \leq 1$ and with $G(0, \xi)=G(1, \xi)=0$ can be written in the form
$$
G(x, \xi)= \begin{cases}y_{1}(x) y_{2}(\xi) / W(\xi) & 0<x<\xi \\ y_{2}(x) y_{1}(\xi) / W(\xi) & \xi<x<1\end{cases}
$$
where $W(x)=W\left[y_{1}(x), y_{2}(x)\right]$ is the Wronskian of $y_{1}(x)$ and $y_{2}(x)$.

Linh Vu
Linh Vu
Numerade Educator
07:14

Problem 28

Use the result of the previous exercise to find the Green's function $G(x, \xi)$ that satisfies
$$
\frac{d^{2} G}{d x^{2}}+3 \frac{d G}{d x}+2 G=\delta(x-x)
$$
in the interval $0 \leq x, \xi \leq 1$ with $G(0, \xi)=G(1, \xi)=0 .$ Hence obtain integral expressions for the solution of
$$
\frac{d^{2} y}{d x^{2}}+3 \frac{d y}{d x}+2 y= \begin{cases}0 & 0<x<x_{0} \\ 1 & x_{0}<x<1\end{cases}
$$
distinguishing between the cases (a) $x<x_{0}$, and (b) $x>x_{0}$.

Keshav Singh
Keshav Singh
Numerade Educator
00:58

Problem 29

The equation of motion for a driven damped harmonic oscillator can be written
$$
\ddot{x}+2 \dot{x}+\left(1+\kappa^{2}\right) x=f(t)
$$
with $\kappa \neq 0$. If it starts from rest with $x(0)=0$ and $\dot{x}(0)=0$, find the corresponding Green's function $G(t, \tau)$ and verify that it can be written as a function of $t-\tau$ only. Find the explicit solution when the driving force is the unit step function, i.e. $f(t)=H(t)$. Confirm your solution by taking the Laplace transforms of both it and the original equation.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:55

Problem 30

Show that the Green's function for the equation
$$
\frac{d^{2} y}{d x^{2}}+\frac{y}{4}=f(x)
$$
subject to the boundary conditions $y(0)=y(\pi)=0$, is given by
$$
G(x, z)= \begin{cases}-2 \cos \frac{1}{2} x \sin \frac{1}{2} z & 0 \leq z \leq x \\ -2 \sin \frac{1}{2} x \cos \frac{1}{2} z & x \leq z \leq \pi.\end{cases}
$$

Keshav Singh
Keshav Singh
Numerade Educator
09:14

Problem 31

Find the Green's function $x=G\left(t, t_{0}\right)$ that solves
$$
\frac{d^{2} x}{d t^{2}}+\alpha \frac{d x}{d t}=\delta\left(t-t_{0}\right)
$$
under the initial conditions $x=d x / d t=0$ at $t=0$. Hence solve
$$
\frac{d^{2} x}{d t^{2}}+\alpha \frac{d x}{d t}=f(t)
$$
where $f(t)=0$ for $t<0$
Evaluate your answer explicitly for $f(t)=A e^{-a t}(t>0)$.

Mike Gaerlan
Mike Gaerlan
Numerade Educator
02:04

Problem 32

(a) By multiplying through by $d y / d x$, write down the solution to the equation
$$
\frac{d^{2} y}{d x^{2}}+f(y)=0
$$
where $f(y)$ can be any function.
(b) A mass $m$, initially at rest at the point $x=0$, is accelerated by a force
$$
f(x)=A\left(x_{0}-x\right)\left[1+2 \ln \left(1-\frac{x}{x_{0}}\right)\right]
$$
Its equation of motion is $m d^{2} x / d t^{2}=f(x) .$ Find $x$ as a function of time and show that ultimately the particle has travelled a distance $x_{0}$.

Dharmendra Jain
Dharmendra Jain
Numerade Educator
07:40

Problem 33

Solve
$$
2 y \frac{d^{3} y}{d x^{3}}+2\left(y+3 \frac{d y}{d x}\right) \frac{d^{2} y}{d x^{2}}+2\left(\frac{d y}{d x}\right)^{2}=\sin x.
$$

Keshav Singh
Keshav Singh
Numerade Educator
02:35

Problem 34

Find the general solution of the equation
$$
x \frac{d^{3} y}{d x^{3}}+2 \frac{d^{2} y}{d x^{2}}=A x.
$$

Keshav Singh
Keshav Singh
Numerade Educator
00:46

Problem 35

Express the equation
$$
\frac{d^{2} y}{d x^{2}}+4 x \frac{d y}{d x}+\left(4 x^{2}+6\right) y=e^{-x^{2}} \sin 2 x
$$
in canonical form and hence find its general solution.

Sajin Shajee
Sajin Shajee
Numerade Educator
06:26

Problem 36

Find the form of the solutions of the equation
$$
\frac{d y}{d x} \frac{d^{3} y}{d x^{3}}-2\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+\left(\frac{d y}{d x}\right)^{2}=0
$$
which have $y(0)=\infty$
(You will need the result $\int^{z} \operatorname{cosech} u d u=-\ln (\operatorname{cosech} z+\operatorname{coth} z)$.)

Keshav Singh
Keshav Singh
Numerade Educator
01:05

Problem 37

Consider the equation
$$
x^{p} y^{\prime \prime}+\frac{n+3-2 p}{n-1} x^{p-1} y^{\prime}+\left(\frac{p-2}{n-1}\right)^{2} x^{p-2} y=y^{n}
$$
in which $p \neq 2$ and $n>-1$ but $n \neq 1$. For the boundary conditions $y(1)=0$ and $y^{\prime}(1)=\lambda$, show that the solution is $y(x)=v(x) x^{(p-2) /(n-1)}$, where $v(x)$ is given by
$$
\int_{0}^{v(x)} \frac{d z}{\left[\lambda^{2}+2 z^{n+1} /(n+1)\right]^{1 / 2}}=\ln x.
$$

Victor Salazar
Victor Salazar
Numerade Educator