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Mathematical Methods for Physics and Engineering: A Comprehensive Guide

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

Chapter 16

Series solutions of ordinary differential equations - all with Video Answers

Educators

Chapter Questions

01:53

Problem 1

Find two power series solutions about $z=0$ of the differential equation
$$
\left(1-z^{2}\right) y^{\prime \prime}-3 z y^{\prime}+\lambda y=0
$$
Deduce that the value of $\lambda$ for which the corresponding power series becomes an $N$ th-degree polynomial $U_{N}(z)$ is $N(N+2)$. Construct $U_{2}(z)$ and $U_{3}(z)$.

Amy Jiang
Amy Jiang
Numerade Educator
01:53

Problem 2

Find solutions, as power series in $z$, of the equation
$$
4 z y^{\prime \prime}+2(1-z) y^{\prime}-y=0.
$$
Identify one of the solutions and verify it by direct substitution.

Amy Jiang
Amy Jiang
Numerade Educator
01:11

Problem 3

Find power series solutions in $z$ of the differential equation
$$
z y^{\prime \prime}-2 y^{\prime}+9 z^{5} y=0.
$$
Identify closed forms for the two series, calculate their Wronskian, and verify that they are linearly independent. Compare the Wronskian with that calculated from the differential equation.

Raj Bala
Raj Bala
Numerade Educator
07:54

Problem 4

Change the independent variable in the equation
$$
\frac{d^{2} f}{d z^{2}}+2(z-a) \frac{d f}{d z}+4 f=0
$$
from $z$ to $x=z-\alpha$, and find two independent series solutions, expanded about $x=0$, of the resulting equation. Deduce that the general solution of $(*)$ is
$$
f(z, \alpha)=A(z-\alpha) e^{-(z-\alpha)^{2}}+B \sum_{m=0}^{\infty} \frac{(-4)^{m} m !}{(2 m) !}(z-\alpha)^{2 m}
$$
with $A$ and $B$ arbitrary constants.

Keshav Singh
Keshav Singh
Numerade Educator
01:05

Problem 5

(a) Verify that $z=1$ is a regular singular point of Legendre's equation and that the indicial equation for a series solution in powers of $(z-1)$ has roots 0 and $3 .$
(b) Obtain the corresponding recurrence relation and show that $\sigma=0$ does not give a valid series solution.
(c) Determine the radius of convergence $R$ of the $\sigma=3$ series and relate it to the positions of the singularities of Legendre's equation.

Raj Bala
Raj Bala
Numerade Educator
03:29

Problem 6

Verify that $z=0$ is a regular singular point of the equation
$$
z^{2} y^{\prime \prime}-\frac{3}{2} z y^{\prime}+(1+z) y=0
$$
and that the indicial equation has roots 2 and $1 / 2 .$ Show that the general solution is
$$
\begin{aligned}
y(z)=& 6 a_{0} z^{2} \sum_{n=0}^{\infty} \frac{(-1)^{n}(n+1) 2^{2 n} z^{n}}{(2 n+3) !} \\
&+b_{0}\left(z^{1 / 2}+2 z^{3 / 2}-\frac{z^{1 / 2}}{4} \sum_{n=2}^{\infty} \frac{(-1)^{n} 2^{2 n} z^{n}}{n(n-1)(2 n-3) !}\right).
\end{aligned}
$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
04:07

Problem 7

Use the derivative method to obtain as a second solution of Bessel's equation for the case when $v=0$ the following expression:
$$
J_{0}(z) \ln z-\sum_{n=1}^{\infty} \frac{(-1)^{n}}{(n !)^{2}}\left(\sum_{r=1}^{n} \frac{1}{r}\right)\left(\frac{z}{2}\right)^{2 n},
$$
given that the first solution is $J_{0}(z)$ as specified by (16.63).

Amy Jiang
Amy Jiang
Numerade Educator
03:24

Problem 8

By initially writing $y(x)$ as $x^{1 / 2} f(x)$ and then making subsequent changes of variable, reduce
$$
\frac{d^{2} y}{d x^{2}}+\lambda x y=0
$$
to Bessel's equation. Hence show that a solution that is finite at $x=0$ is a multiple of $x^{1 / 2} J_{1 / 3}\left(\frac{2}{3} \sqrt{\lambda x^{3}}\right)$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
08:37

Problem 9

(a) Show that the indicial equation for
$$
z y^{\prime \prime}-2 y^{\prime}+y z=0
$$
has roots that differ by an integer but that the two roots nevertheless generate linearly independent solutions
$$
\begin{aligned}
&y_{1}(z)=3 a_{0} \sum_{n=1}^{\infty} \frac{(-1)^{n+1} 2 n z^{2 n+1}}{(2 n+1) !} \\
&y_{2}(z)=a_{0} \sum_{n=0}^{\infty} \frac{(-1)^{n+1}(2 n-1) z^{2 n}}{(2 n) !}
\end{aligned}
$$
(b) Show that $y_{1}(z)$ is equal to $3 a_{0}(\sin z-z \cos z)$ by expanding the sinusoidal functions. Then, using the Wronskian method, find an expression for $y_{2}(z)$ in terms of sinusoids. (You will need to write $z^{2}$ as $(z / \sin z)(z \sin z)$ and integrate by parts to evaluate the integral involved.)
(c) Confirm that the two solutions are linearly independent by showing that their Wronskian is equal to $-z^{2}$, in accordance with (16.4).

Mike Gaerlan
Mike Gaerlan
Numerade Educator
01:45

Problem 10

Find series solutions of the equation $y^{\prime \prime}-2 z y^{\prime}-2 y=0$. Identify one of the series as $y_{1}(z)=\exp z^{2}$ and verify this by direct substitution. By setting $y_{2}(z)=u(z) y_{1}(z)$ and solving the resulting equation for $u(z)$, find an explicit form for $y_{2}(z)$ and deduce that
$$
\int_{0}^{x} e^{-v^{2}} d v=e^{-x^{2}} \sum_{n=0}^{\infty} \frac{n !}{2(2 n+1) !}(2 x)^{2 n+1}.
$$

James Kiss
James Kiss
Numerade Educator
01:29

Problem 11

(a) Identify and classify the singular points of the equation
$$
z(1-z) \frac{d^{2} y}{d z^{2}}+(1-z) \frac{d y}{d z}+\lambda y=0
$$
and determine their indices.
(b) Find one series solution in powers of $z$. Give a formal expression for a second linearly independent solution.
(c) Deduce the values of $\lambda$ for which there is a polynomial solution $P_{N}(z)$ of degree $N$. Evaluate the first four polynomials, normalised in such a way that $P_{N}(0)=1$.

AP
Andreas Papavassiliou
Numerade Educator
01:47

Problem 12

Find the general power series solution about $z=0$ of the equation
$$
z \frac{d^{2} y}{d z^{2}}+(2 z-3) \frac{d y}{d z}+\frac{4}{z} y=0.
$$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:11

Problem 13

Find the radius of convergence of a series solution about the origin for the equation $\left(z^{2}+a z+b\right) y^{\prime \prime}+2 y=0$ in the following cases:
(a) $a=5, b=6 ;$ (b) $a=5, b=7$.
Show that if $a$ and $b$ are real and $4 b>a^{2}$ then the radius of convergence is always given by $b^{1 / 2}$.

Linda Hand
Linda Hand
Numerade Educator
02:14

Problem 14

For the equation $y^{\prime \prime}+z^{-3} y=0$, show that the origin becomes a regular singular point if the independent variable is changed from $z$ to $x=1 / z$. Hence find a series solution of the form $y_{1}(z)=\sum_{0}^{\infty} a_{n} z^{-n}$. By setting $y_{2}(z)=u(z) y_{1}(z)$ and expanding the resulting expression for $d u / d z$ in powers of $z^{-1}$, show that $y_{2}(z)$ has the asymptotic form
$$
y_{2}(z)=c\left[z+\ln z-\frac{1}{2}+\mathrm{O}\left(\frac{\ln z}{z}\right)\right]
$$
where $c$ is an arbitrary constant.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
33:09

Problem 15

Prove that the Laguerre equation
$$
z \frac{d^{2} y}{d z^{2}}+(1-z) \frac{d y}{d z}+\lambda y=0
$$
has polynomial solutions $L_{N}(z)$ if $\lambda$ is a non-negative integer $N$, and determine the recurrence relationship for the polynomial coefficients. Hence show that an expression for $L_{N}(z)$, normalised in such a way that $L_{N}(0)=N !$, is
$$
L_{N}(z)=\sum_{n=0}^{N} \frac{(-1)^{n}(N !)^{2}}{(N-n) !(n !)^{2}} z^{n}
$$
Evaluate $L_{3}(z)$ explicitly. [The Laguerre generating function is discussed in exercise 17.9.]

Lucas Finney
Lucas Finney
Numerade Educator
18:46

Problem 16

(a) Use Leibniz' theorem to show that the Rodrigues' formula for the Laguerre polynomials $L_{N}(z)$ of the previous question is
$$
L_{N}(z)=e^{z} \frac{d^{N}}{d z^{N}}\left(z^{N} e^{-z}\right)
$$
(b) Use the Rodrigue formulation to prove that
$$
z L_{N}^{\prime}(z)=L_{N+1}(z)-(N+1-z) L_{N}(z)
$$
(c) Deduce the recurrence relation for the Laguerre polynomials, namely
$$
L_{N+1}(z)+(z-2 N-1) L_{N}(z)+N^{2} L_{N-1}(z)=0.
$$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:42

Problem 17

Equation (16.32) was shown to have a polynomial solution provided that $\lambda=2 n$ with $n$ an integer $\geq 0$. The polynomials are known as Hermite polynomials $H_{n}(x)$ and are of importance in the quantum mechanical treatment of the harmonic oscillator problem. They may also be defined by
$$
\Phi(x, h)=\exp \left(2 x h-h^{2}\right)=\sum_{n=0}^{\infty} \frac{1}{n !} H_{n}(x) h^{n}
$$
Show that
$$
\frac{\partial^{2} \Phi}{\partial x^{2}}-2 x \frac{\partial \Phi}{\partial x}+2 h \frac{\partial \Phi}{\partial h}=0
$$
and hence that the $H_{n}(x)$ satisfy (16.32). Use $\Phi$ to prove that
(a) $H_{n}^{\prime}(x)=2 n H_{n-1}(x)$,
(b) $H_{n+1}^{n}(x)-2 x H_{n}(x)+2 n H_{n-1}(x)=0 .$

Brian Ketelobeter
Brian Ketelobeter
Numerade Educator
01:07

Problem 18

By writing $\Phi(x, h)$ of the previous exercise as a function of $h-x$ rather than of $h$, show that an alternative representation of the $n$th Hermite polynomial is
$$
H_{n}(x)=(-1)^{n}\left(\exp x^{2}\right) \frac{d^{n}}{d x^{n}}\left[\exp \left(-x^{2}\right)\right]
$$
(Note that $H_{n}(x)=\partial^{n} \Phi / \partial h^{n}$ at $h=0 .$ )

Clarissa Noh
Clarissa Noh
Numerade Educator
01:20

Problem 19

Obtain the recurrence relations for the solution of Legendre's equation (16.35) in inverse powers of $z$, i.e. set $y(z)=\sum a_{n} z^{\sigma-n}$, with $a_{0} \neq 0 .$ Deduce that if $\ell$ is an integer then the series with $\sigma=\ell$ will terminate and hence converge for all $z$ whilst that with $\sigma=-(\ell+1)$ does not terminate and hence converges only for $|z|>1$.

James Kiss
James Kiss
Numerade Educator
View

Problem 20

Carry through the following procedure as an alternative proof of result (16.45).
(a) Square both sides of (16.49), giving the generating-function definition of the Legendre polynomials.
(b) Express the RHS as a sum of powers of $h$, obtaining expressions for the coefficients.
(c) Integrate the RHS from $-1$ to 1 and use the orthogonality results (16.46).
(d) Similarly integrate the LHS and expand the result in powers of $h$.
(e) Compare coefficients.

Victor Salazar
Victor Salazar
Numerade Educator
07:28

Problem 21

A charge $+2 q$ is situated at the origin and charges of $-q$ are situated at distances $\pm a$ from it along the polar axis. By relating it to the generating function for the Legendre polynomials, show that the electrostatic potential $\Phi$ at a point $(r, \theta, \phi)$ with $r>a$ is given by
$$
\Phi(r, \theta, \phi)=\frac{2 q}{4 \pi \epsilon_{0} r} \sum_{s=1}^{\infty}\left(\frac{a}{r}\right)^{2 s} P_{2 s}(\cos \theta).
$$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:07

Problem 22

The origin is an ordinary point of the Chebyshev equation,
$$
\left(1-z^{2}\right) y^{\prime \prime}-z y^{\prime}+m^{2} y=0
$$
which therefore has series solutions of the form $z^{\sigma} \sum_{0}^{\infty} a_{n} z^{n}$ for $\sigma=0$ and $\sigma=1$.
(a) Find the recurrence relationships for the $a_{n}$ in the two cases and show that there exist polynomial solutions $T_{m}(z)$ :
(i) for $\sigma=0$, when $m$ is an even integer, the polynomial having $\frac{1}{2}(m+2)$ terms;
(ii) for $\sigma=1$, when $m$ is an odd integer, the polynomial having $\frac{1}{2}(m+1)$ terms.
(b) $T_{m}(z)$ is normalised so as to have $T_{m}(1)=1 .$ Find explicit forms for $T_{m}(z)$ for $m=0,1,2,3$.
(c) Show that the corresponding non-terminating series solutions $S_{m}(z)$ have as their first few terms
$$
\begin{aligned}
&S_{0}(z)=a_{0}\left(z+\frac{1}{3 !} z^{3}+\frac{9}{5 !} z^{5}+\cdots\right) \\
&S_{1}(z)=a_{0}\left(1-\frac{1}{2 !} z^{2}-\frac{3}{4 !} z^{4}-\cdots\right) \\
&S_{2}(z)=a_{0}\left(z-\frac{3}{3 !} z^{3}-\frac{15}{5 !} z^{5}-\cdots\right) \\
&S_{3}(z)=a_{0}\left(1-\frac{9}{2 !} z^{2}+\frac{45}{4 !} z^{4}+\cdots\right)
\end{aligned}
$$

Raj Bala
Raj Bala
Numerade Educator
05:21

Problem 23

By choosing a suitable form for $h$ in $(16.82)$, show that further integral repesentations of the Bessel functions of the first kind are given, for integral $m$, by
$$
\begin{aligned}
J_{2 m}(z) &=\frac{(-1)^{m}}{\pi} \int_{0}^{2 \pi} \cos (z \cos \theta) \cos 2 m \theta d \theta & m \geq 1 \\
J_{2 m+1}(z) &=\frac{(-1)^{m+1}}{\pi} \int_{0}^{2 \pi} \cos (z \cos \theta) \sin (2 m+1) \theta d \theta & m \geq 0.
\end{aligned}
$$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
06:25

Problem 24

Show from the definition given in (16.66) that the Bessel function of the second kind of order $v$ can be written as
$$
Y_{v}(z)=\frac{1}{\pi}\left[\frac{\partial J_{\mu}(z)}{\partial \mu}-(-1)^{v} \frac{\partial J_{-\mu}(z)}{\partial \mu}\right]_{\mu=v}
$$
Using the explicit expression (16.63) for $J_{\mu}(z)$, show that $\partial J_{\mu}(z) / \partial \mu$ can be written as
$$
J_{v}(z) \ln \left(\frac{z}{2}\right)+g(v, z)
$$
and deduce that $Y_{v}(z)$ can be expressed as
$$
Y_{v}(z)=\frac{2}{\pi} J_{v}(z) \ln \left(\frac{z}{2}\right)+h(v, z)
$$
$h(v, z)$, like $g(v, z)$, being a power series in $z$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator