Mathematical Methods for Physics and Engineering: A Comprehensive Guide

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

Chapter 18 Partial differential equations: general and particular solutions - all with Video Answers

Educators

Chapter Questions

Problem 1

Determine whether the following can be written as functions of $p=x^{2}+2 y$ only, and hence whether they are solutions of $(18.8)$ :
(a) $x^{2}\left(x^{2}-4\right)+4 y\left(x^{2}-2\right)+4\left(y^{2}-1\right)$;
(b) $x^{4}+2 x^{2} y+y^{2}$;
(c) $\left[x^{4}+4 x^{2} y+4 y^{2}+4\right] /\left[2 x^{4}+x^{2}(8 y+1)+8 y^{2}+2 y\right]$.

Willis James

Numerade Educator

01:14

Problem 2

Find partial differential equations satisfied by the following functions $u(x, y)$ for all arbitrary functions $f$ and all arbitrary constants $a$ and $b$ :
(a) $u(x, y)=f\left(x^{2}-y^{2}\right)$
(b) $u(x, y)=(x-a)^{2}+(y-b)^{2}$;
(c) $u(x, y)=y^{n} f(y / x)$;
(d) $u(x, y)=f(x+a y)$.

Lucas Finney

Numerade Educator

01:58

Problem 3

Solve the following partial differential equations for $u(x, y)$ with the boundary conditions given:
(a) $x \frac{\partial u}{\partial x}+x y=u, \quad u=2 y$ on the line $x=1$
(b) $1+x \frac{\partial u}{\partial y}=x u, \quad u(x, 0)=x$.

Gregory Higby

Numerade Educator

09:28

Problem 4

Find the most general solutions $u(x, y)$ of the following equations consistent with the boundary conditions stated:
(a) $y \frac{\partial u}{\partial x}-x \frac{\partial u}{\partial y}=0, \quad u(x, 0)=1+\sin x$
(b) $i \frac{\partial u}{\partial x}=3 \frac{\partial u}{\partial y}, \quad u=(4+3 i) x^{2}$ on the line $x=y$;
(c) $\sin x \sin y \frac{\partial u}{\partial x}+\cos x \cos y \frac{\partial u}{\partial y}=0, \quad u=\cos 2 y$ on $x+y=\pi / 2$
(d) $\frac{\partial u}{\partial x}+2 x \frac{\partial u}{\partial y}=0, \quad u=2$ on the parabola $y=x^{2}$.

Sajin Shajee

Numerade Educator

02:13

Problem 5

Find solutions of
$$
\frac{1}{x} \frac{\partial u}{\partial x}+\frac{1}{y} \frac{\partial u}{\partial y}=0
$$
for which (a) $u(0, y)=y$, (b) $u(1,1)=1$.

Manik Pulyani

Numerade Educator

02:24

Problem 6

Find the most general solutions $u(x, y)$ of the following equations consistent with the boundary conditions stated:
(a) $y \frac{\partial u}{\partial x}-x \frac{\partial u}{\partial y}=3 x, \quad u=x^{2}$ on the line $y=0$; (b) $y \frac{\partial u}{\partial x}-x \frac{\partial u}{\partial y}=3 x, \quad u(1,0)=2$;
(c) $y^{2} \frac{\partial u}{\partial x}+x^{2} \frac{\partial u}{\partial y}=x^{2} y^{2}\left(x^{3}+y^{3}\right)$, no boundary conditions.

Uma Kumari

Numerade Educator

07:07

Problem 7

Solve
$$
\sin x \frac{\partial u}{\partial x}+\cos x \frac{\partial u}{\partial y}=\cos x
$$
subject to (a) $u(\pi / 2, y)=0$, (b) $u(\pi / 2, y)=y(y+1)$.

Khoobchandra Agrawal

Numerade Educator

01:11

Problem 8

A function $u(x, y)$ satisfies
$$
2 \frac{\partial u}{\partial x}+3 \frac{\partial u}{\partial y}=10
$$
and takes the value 3 on the line $y=4 x$. Evaluate $u(2,4)$.

Will Erickson

Numerade Educator

05:18

Problem 9

If $u(x, y)$ satisfies
$$
\frac{\partial^{2} u}{\partial x^{2}}-3 \frac{\partial^{2} u}{\partial x \partial y}+2 \frac{\partial^{2} u}{\partial y^{2}}=0
$$
and $u=-x^{2}$ and $\partial u / \partial y=0$ for $y=0$ and all $x$, find the value of $u(0,1)$.

Aman Gupta

Numerade Educator

01:55

Problem 10

(a) Solve the previous question if the boundary condition is $u=\partial u / \partial y=1$ when $y=0$ for all $x$.
(b) In which region of the $x y$-plane would $u$ be determined if the boundary condition were $u=\partial u / \partial y=1$ when $y=0$ for all $x>0$ ?

Hast Aggarwal

Numerade Educator

03:39

Problem 11

In those cases in which it is possible to do so, evaluate $u(2,2)$, where $u(x, y)$ is the solution of
$$
2 y \frac{\partial u}{\partial x}-x \frac{\partial u}{\partial y}=2 x y\left(2 y^{2}-x^{2}\right)
$$
that satisfies the (separate) boundary conditions given below.
(a) $u(x, 1)=x^{2}$ for all $x$.
(b) $u(x, 1)=x^{2}$ for $x \geq 0$.
(c) $u(x, 1)=x^{2}$ for $0 \leq x \leq 3$
(d) $u(x, 0)=x$ for $x \geq 0$
(e) $u(x, 0)=x$ for all $x$.
(f) $u(1, \sqrt{10})=5$
(g) $u(\sqrt{10}, 1)=5$.

Subham Jyoti Mishra

Numerade Educator

03:23

Problem 12

Solve
$$
6 \frac{\partial^{2} u}{\partial x^{2}}-5 \frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^{2} u}{\partial y^{2}}=14
$$
subject to $u=2 x+1$ and $\partial u / \partial y=4-6 x$, both on the line $y=0$.

Sanchit Jain

Numerade Educator

01:54

Problem 13

By changing the independent variables in the previous question to
$$
\xi=x+2 y \quad \text { and } \quad \eta=x+3 y
$$
show that it must be possible to write $14\left(x^{2}+5 x y+6 y^{2}\right)$ in the form
$$
f_{1}(x+2 y)+f_{2}(x+3 y)-\left(x^{2}+y^{2}\right)
$$
and determine the forms of $f_{1}(z)$ and $f_{2}(z)$.

Manik Pulyani

Numerade Educator

02:00

Problem 14

Solve
$$
\frac{\partial^{2} u}{\partial x \partial y}+3 \frac{\partial^{2} u}{\partial y^{2}}=x(2 y+3 x)
$$.

Khoobchandra Agrawal

Numerade Educator

02:00

Problem 15

Find the most general solution of $\partial^{2} u / \partial x^{2}+\partial^{2} u / \partial y^{2}=x^{2} y^{2}$.

Khoobchandra Agrawal

Numerade Educator

01:14

Problem 16

An infinitely long string on which waves travel at speed $c$ has an initial displacement
$$
y(x)= \begin{cases}\sin (\pi x / a), & -a \leq x \leq a \\ 0, & |x|>a\end{cases}
$$
It is released from rest at time $t=0$, and its subsequent displacement is described by $y(x, t)$.

By expressing the initial displacement as one explicit function incorporating Heaviside step functions, find an expression for $y(x, t)$ at a general time $t>0$. In particular, determine the displacement as a function of time (a) at $x=0$, (b) at $x=a$, and (c) at $x=a / 2$.

Hast Aggarwal

Numerade Educator

03:50

Problem 17

The non-relativistic Schrödinger equation (18.7) is similar to the diffusion equation in having different orders of derivatives in its various terms; this precludes solutions that are arbitrary functions of particular linear combinations of variables. However, since exponential functions do not change their forms under differentiation, solutions in the form of exponential functions of combinations of the variables may still be possible.
Consider the Schrödinger equation for the case of a constant potential, i.e. for a free particle, and show that it has solutions of the form $A \exp (l x+m y+n z+\lambda t$ ) where the only requirement is that $$
-\frac{\hbar^{2}}{2 m}\left(l^{2}+m^{2}+n^{2}\right)=i \hbar \lambda
$$ In particular, identify the equation and wavefunction obtained by taking $\lambda$ as $-i E / \hbar$, and $l, m$ and $n$ as $i p_{x} / \hbar, i p_{y} / \hbar$ and $i p_{z} / \hbar$ respectively, where $E$ is the energy and $p$ the momentum of the particle; these identifications are essentially the content of the de Broglie and Einstein relationships.

Laszlo Zalavari

Numerade Educator

03:33

Problem 18

Like the Schrödinger equation of the previous question, the equation describing the transverse vibrations of a rod,
$$
a^{4} \frac{\partial^{4} u}{\partial x^{4}}+\frac{\partial^{2} u}{\partial t^{2}}=0
$$
has different orders of derivatives in its various terms. Show, however, that it has solutions of exponential form $u(x, t)=A \exp (\lambda x+i \omega t)$ provided that the relation $a^{4} \lambda^{4}=\omega^{2}$ is satisfied.

Use a linear combination of such allowed solutions, expressed as the sum of sinusoids and hyperbolic sinusoids of $\lambda x$, to describe the transverse vibrations of a rod of length $L$ clamped at both ends. At a clamped point both $u$ and $\partial u / \partial x$ must vanish; show that this implies that $\cos (\lambda L) \cosh (\lambda L)=1$, thus determining the frequencies $\omega$ at which the rod can vibrate.

Khoobchandra Agrawal

Numerade Educator

08:36

Problem 19

An incompressible fluid of density $\rho$ and negligible viscosity flows with velocity $v$ along a thin straight tube, perfectly light and flexible, of cross-section $A$ and held under tension $T$. Assume that small transverse displacements $u$ of the tube are governed by
$$
\frac{\partial^{2} u}{\partial t^{2}}+2 v \frac{\partial^{2} u}{\partial x \partial t}+\left(v^{2}-\frac{T}{\rho A}\right) \frac{\partial^{2} u}{\partial x^{2}}=0
$$
(a) Show that the general solution consists of a superposition of two waveforms travelling with different speeds.
(b) The tube initially has a small transverse displacement $u=a \cos k x$ and is suddenly released from rest. Find its subsequent motion.

Sheh Lit Chang

University of Washington

02:00

Problem 20

A sheet of material of thickness $w$, specific heat capacity $c$ and thermal conductivity $k$ is isolated in a vacuum, but its two sides are exposed to fluxes of radiant heat of strengths $J_{1}$ and $J_{2}$. Ignoring short-term transients, show that the temperature difference between its two surfaces is steady at $\left(J_{2}-J_{1}\right) w / 2 k$, whilst their average temperature increases at a rate $\left(J_{2}+J_{1}\right) / c w$.

Mahendra K

Numerade Educator

00:56

Problem 21

In an electrical cable of resistance $R$ and capacitance $C$ per unit length, voltage signals obey the equation $\partial^{2} V / \partial x^{2}=R C \partial V / \partial t$. This has solutions of the form given in (18.36) and also of the form $V=A x+D$
(a) Find a combination of these that represents the situation after a steady voltage $V_{0}$ is applied at $x=0$ at time $t=0$.
(b) Obtain a solution describing the propagation of the voltage signal resulting from application of the signal $V=V_{0}$ for $0<t<T, V=0$ otherwise, to the end $x=0$ of an infinite cable.
(c) Show that for $t \gg T$ the maximum signal occurs at a value of $x$ proportional to $t^{1 / 2}$ and has a magnitude proportional to $t^{-1}$.

Manik Pulyani

Numerade Educator

02:29

Problem 22

The daily and annual variations of temperature at the surface of the earth may be represented by sine-wave oscillations with equal amplitudes and periods of 1 day and 365 days respectively. Assume that for (angular) frequency $\omega$ the temperature at depth $x$ in the earth is given by $u(x, t)=A \sin (\omega t+\mu x) \exp (-\lambda x)$, where $\lambda$ and $\mu$ are constants.
(a) Use the diffusion equation to find the values of $\lambda$ and $\mu$.
(b) Find the ratio of the depths below the surface at which the amplitudes have dropped to $1 / 20$ of their surface values.
(c) At what time of year is the soil coldest at the greater of these depths, assuming that the smoothed annual variation in temperature at the surface has a minimum on February $1 \mathrm{st}$ ?

Suzanne W.

Numerade Educator

11:44

Problem 23

Consider each of the following situations in a qualitative way and determine the equation type, the nature of the boundary curve and the type of boundary conditions involved.
(a) a conducting bar given an initial temperature distribution and then thermally isolated;
(b) two long conducting concentric cylinders on each of which the voltage distribution is specified;
(c) two long conducting concentric cylinders on each of which the charge distribution is specified;
(d) a semi-infinite string the end of which is made to move in a prescribed way.

Nathan Silvano

Numerade Educator

00:53

Problem 24

This example gives a formal demonstration that the type of a second-order PDE (elliptic, parabolic or hyperbolic) cannot be changed by a new choice of independent variable. The algebra is somewhat lengthy, but straightforward.

If a change of variable $\xi=\xi(x, y), \eta=\eta(x, y)$ is made in (18.19), so that it reads
$$
A^{\prime} \frac{\partial^{2} u}{\partial \xi^{2}}+B^{\prime} \frac{\partial^{2} u}{\partial \xi \partial \eta}+C^{\prime} \frac{\partial^{2} u}{\partial \eta^{2}}+D^{\prime} \frac{\partial u}{\partial \xi}+E^{\prime} \frac{\partial u}{\partial \eta}+F^{\prime} u=R^{\prime}(\xi, \eta)
$$
show that
$$
B^{\prime 2}-4 A^{\prime} C^{\prime}=\left(B^{2}-4 A C\right)\left[\frac{\partial(\xi, \eta)}{\partial(x, y)}\right]^{2}
$$
Hence deduce the conclusion stated above.

Hast Aggarwal

Numerade Educator

03:50

Problem 25

The Klein-Gordon equation (which is satisfied by the quantum-mechanical wavefunction $\Phi(\mathbf{r})$ of a relativistic spinless particle of non-zero mass $m$ ) is
$$
\nabla^{2} \Phi-m^{2} \Phi=0
$$.

Laszlo Zalavari

Numerade Educator