Chapter 11

Determine two linearly independent power series solutions to the given differential equation centered at \(x=0 .\) Give a lower bound on the radius of convergence of the series solutions obtained. $$\left(x^{2}-3\right) y^{\prime \prime}-3 x y^{\prime}-5 y=0.$$

Determine the radius of convergence of the power series representation of the given function with center \(x_{0}\). $$f(x)=\frac{2 x}{x^{2}+16}, \quad x_{0}=1$$

By manipulating the general expression for \(J_{1 / 2}(x)\) show that it can be written in closed form as $$ J_{1 / 2}(x)=\sqrt{\frac{2}{\pi x}} \sin x $$

Determine whether \(x=0\) is an ordinary point or a regular singular point of the given differential equation. Then obtain two linearly independent solutions to the differential equation and state the maximum interval on which your solutions are valid. $$4 x y^{\prime \prime}+3 y^{\prime}+3 y=0.$$

Show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on \((0, \infty)\). $$4 x^{2} y^{\prime \prime}+3 x y^{\prime}+x y=0$$

The indicial equation and recurrence relation for the differential equation $$x^{2} y^{\prime \prime}+x[(2-b)+x] y^{\prime}-(b-\gamma x) y=0 $$are, respectively,$$ \begin{aligned} (r+1)(r-b) &=0 \\ (r+n+1)(r+n-b) a_{n} &=-[(r+n-1)+\gamma] a_{n-1} \\ n &=1,2,3, \ldots \end{aligned}$$ in the usual notation, where \(b\) and \(\gamma\) are constants. Determine the form of two linearly independent series solutions to Equation \((11.5 .37)\) on (a) \(b\) not an integer. (b) \(b=-1\) (c) \(b=N,\) a nonnegative integer. [For solutions containing a term of the form \(A y_{1}(x) \ln x,\) you must determine whether \(A\) is zero or nonzero.]\((0, \infty)\) in the following cases:

Given that $$ J_{1 / 2}(x)=\sqrt{\frac{2}{\pi x}} \sin x, \quad J_{-1 / 2}(x)=\sqrt{\frac{2}{\pi x}} \cos x $$ express \(J_{3 / 2}(x)\) and \(J_{-3 / 2}(x)\) in closed form. Convince yourself that all half-integer order Bessel functions of the first kind can be expressed as a finite sum of terms involving products of \(\sin x, \cos x,\) and powers of \(x\).

Determine the radius of convergence of the power series representation of the given function with center \(x_{0}\). $$f(x)=\frac{x^{2}-3}{x^{2}-2 x+5}, \quad x_{0}=0$$

Deal with Hermite's equation: $$y^{\prime \prime}-2 x y^{\prime}+2 \alpha y=0, \quad-\infty

Determine two linearly independent power series solutions to the given differential equation centered at \(x=0 .\) Give a lower bound on the radius of convergence of the series solutions obtained. $$\left(1+x^{2}\right) y^{\prime \prime}+4 x y^{\prime}+2 y=0.$$

Show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on \((0, \infty)\). $$6 x^{2} y^{\prime \prime}+x(1+18 x) y^{\prime}+(1+12 x) y=0$$

By integrating the recurrence relation for derivatives of the Bessel functions of the first kind, show that (a) \(\int x^{p} J_{p-1}(x) d x=x^{p} J_{p}(x)+C\) (b) \(\int x^{-p} J_{p+1}(x) d x=-x^{-p} J_{p}(x)+C\)

Determine two linearly independent power series solutions to the given differential equation centered at \(x=0 .\) Give a lower bound on the radius of convergence of the series solutions obtained. $$\left(1-4 x^{2}\right) y^{\prime \prime}-20 x y^{\prime}-16 y=0.$$

Determine whether \(x=0\) is an ordinary point or a regular singular point of the given differential equation. Then obtain two linearly independent solutions to the differential equation and state the maximum interval on which your solutions are valid. $$x^{2} y^{\prime \prime}+\frac{3}{2} x y^{\prime}-\frac{1}{2}(1+x) y=0.$$

Show that $$x^{2}(1+x) y^{\prime \prime}+x^{2} y^{\prime}-2 y=0$$ has two linearly independent Frobenius series solutions on (-1,1) and find them.

\(\diamond\) Use some form of technology to determine the coefficients in the Legendre expansion of the polynomial \(p(x)=3 x^{3}-1\).

Determine the radius of convergence of the power series representation of the given function with center \(x_{0}\). $$f(x)=\frac{x}{\left(x^{2}+4 x+13\right)(x-3)}, \quad x_{0}=-1$$

Consider the differential equation \(x^{2} y^{\prime \prime}+3 x y^{\prime}+(1-x) y=0, \quad x>0\) (a) Determine the indicial equation and show that it has a repeated root \(r=-1\) (b) Obtain the corresponding Frobenius series solution. (c) It follows from Theorem 11.5 .1 that Equation \((11.5 .38)\) has a second linearly independent solution of the form $$y_{2}(x)=y_{1}(x) \ln x+\frac{1}{x} \sum_{n=1}^{\infty} b_{n} x^{n}$$ Show that \(b_{1}=-2\) and that in general, $$b_{n}=\frac{1}{n^{2}}\left[b_{n-1}-\frac{2 n}{n !(n-1) !}\right], \quad n=2,3,4, \ldots$$ Use this to find the first three terms of \(y_{2}\)

Show that (a) \(J_{0}^{\prime \prime}(x)=-J_{0}(x)-x^{-1} J_{0}^{\prime}(x)\) (b) \(J_{0}^{\prime \prime \prime}(x)=x^{-1} J_{0}(x)+x^{-2}\left(2-x^{2}\right) J_{0}^{\prime}(x)\).

Determine two linearly independent power series solutions to the given differential equation centered at \(x=0 .\) Give a lower bound on the radius of convergence of the series solutions obtained. $$\left(x^{2}-1\right) y^{\prime \prime}-6 x y^{\prime}+12 y=0.$$

(a) Determine all values of \(x\) at which the function \(f(x)=\frac{1}{x^{2}-1}\) is analytic. (b) Determine the radius of convergence of a power series representation of the function \((11.1 .6)\) centered at \(x=x_{0} .\) (You will need to consider the cases \(-11\) separately.

Determine whether \(x=0\) is an ordinary point or a regular singular point of the given differential equation. Then obtain two linearly independent solutions to the differential equation and state the maximum interval on which your solutions are valid. $$x^{2} y^{\prime \prime}-x(2-x) y^{\prime}+\left(2+x^{2}\right) y=0.$$

Show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on \((0, \infty)\). $$x^{2} y^{\prime \prime}+x y^{\prime}-(2+x) y=0$$

Consider the differential equation $$x y^{\prime \prime}-y=0, \quad x>0$$ (a) Show that the roots of the indicial equation are \(r_{1}=1\) and \(r_{2}=0,\) and determine the Frobenius series solutions corresponding to \(r_{1}=1\) (b) Show that there does not exist a second linearly independent Frobenius series solution. (c) According to Theorem \(11.5 .1,\) Equation \((11.5 .39)\) has a second linearly independent solution of the form $$y_{2}(x)=A y_{1}(x) \ln x+\sum_{n=0}^{\infty} b_{n} x^{n}$$ Show that \(A=b_{0},\) and determine the first three terms in \(y_{2}\)

Show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on \((0, \infty)\). $$2 x y^{\prime \prime}+y^{\prime}-2 x y=0$$

By redefining the ranges of the summations appearing on the left-hand side, show that $$ \begin{aligned} \sum_{n=2}^{\infty} n(n-1) a_{n-1} x^{n-2} &+\sum_{n=1}^{\infty} n a_{n} x^{n-1} \\ &=\sum_{n=0}^{\infty}(n+1)(n+3) a_{n+1} x^{n} \end{aligned} $$

Determine whether \(x=0\) is an ordinary point or a regular singular point of the given differential equation. Then obtain two linearly independent solutions to the differential equation and state the maximum interval on which your solutions are valid. $$x^{2} y^{\prime \prime}-3 x y^{\prime}+4(x+1) y=0.$$

Show that $$ J_{4}(x)=8 x^{-3}\left(6-x^{2}\right) J_{1}(x)-x^{-2}\left(24-x^{2}\right) J_{0}(x) $$

Consider the hypergeometric equation $$x(1-x) y^{\prime \prime}+[c-(a+b+1) x] y^{\prime}-a b y=0,$$ where \(a, b, c\) are constants. (a) Verify that \(x=0\) is a regular singular point of \((11.7 .8)\) (b) Verify that the roots of the indicial equation are \(r_{1}=0, r_{2}=1-c\) (c) Show that the coefficients in a series solution to Equation (11.7.8) centered at \(x=0\) must satisfy the recurrence relation: $$a_{n+1}=\frac{(n+r+a)(n+r+b)}{(n+r+1)(n+r+c)} a_{n},$$ $$n=0,1, \ldots$$ (d) Assuming that \(c\) is not an integer, derive the following solutions to (11.7.8): $$\begin{aligned} &y_{1}(x)=F(a, b, c ; x)=1+\frac{a b}{c} x+\frac{a(a+1) b(b+1)}{c(c+1)} \frac{x^{2}}{2 !}\\\ &+\frac{a(a+1) \cdots(a+n-1) b(b+1) \cdots(b+n-1)}{(c+1) \cdots(c+n-1)} \frac{x^{n}}{n !}\\\ &+\cdots, y_{2}(x)=x^{1-c} F(a+1-c, b+1-c, 2-c ; x). \end{aligned}$$

Determine two linearly independent solutions to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}+x\left(6+x^{2}\right) y^{\prime}+6 y=0$$

Show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on \((0, \infty)\). $$3 x^{2} y^{\prime \prime}-x(x+8) y^{\prime}+6 y=0$$

Determine terms up to and including \(x^{5}\) in two linearly independent power series solutions of the given differential equation. State the radius of convergence of the series solutions. $$y^{\prime \prime}+x y^{\prime}+(2+x) y=0.$$

If \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n},\) where the coefficients in the expansion satisfy $$ \sum_{n=0}^{\infty} n(n+2) a_{n} x^{n}+\sum_{n=1}^{\infty}(n-3) a_{n-1} x^{n}=0 $$ determine \(f(x)\)

Consider the differential equation $$\left(x^{2}-1\right) y^{\prime \prime}+[1-(a+b)] x y^{\prime}+a b y=0, \quad (11.7.9)$$ where \(a\) and \(b\) are constants. (a) Show that the coefficients in a series solution to Equation (11.7.9) centered at \(x=0\) must satisfy the recurrence relation $$a_{n+2}=\frac{(n-a)(n-b)}{(n+2)(n+1)} a_{n}, \quad n=0,1, \ldots$$ and determine two linearly independent series solutions. (b) Show that if either \(a\) or \(b\) is a nonnegative integer, then one of the solutions obtained in (a) is a polynomial. (c) Show that if \(a\) is an odd positive integer and \(b\) is an even positive integer, then both of the solutions defined in (a) are polynomials. (d) If \(a=5\) and \(b=4,\) determine two linearly independent polynomial solutions to Equation (11.7.9). Notice that in this case the radius of convergence of the solutions obtained is \(R=\infty\) whereas Theorem 11.2 .1 only guarantees a radius of convergence \(R \geq 1.\)

Determine two linearly independent solutions to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}+x(1-x) y^{\prime}-y=0$$

Show that (a) \(J_{2}(x)=J_{0}(x)+2 J_{0}^{\prime \prime}(x)\) (b) \(J_{3}(x)=3 J_{1}(x)+4 J_{1}^{\prime \prime}(x)\).

Suppose it is known that the coefficients in the expansion $$ f(x)=\sum_{\Sigma=0}^{x} a_{x} x^{r} $$ satisfy $$ \sum_{n=0}^{\infty}(n+2) a_{n+1} x^{n}-\sum_{n=0}^{\infty} a_{n} x^{n}=0 $$ Show that $$ f(x)=\frac{a_{0}}{x} \sum_{n=0}^{\infty} \frac{1}{(n+1) !} x^{n+1} $$ and express this in terms of familiar elementary functions.

Show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on \((0, \infty)\). $$2 x^{2} y^{\prime \prime}-x(1+2 x) y^{\prime}+2(4 x-1) y=0$$

Determine terms up to and including \(x^{5}\) in two linearly independent power series solutions of the given differential equation. State the radius of convergence of the series solutions. $$y^{\prime \prime}-e^{x} y=0 . \text { [Hint: } e^{x}=1+x+\frac{1}{2 !} x^{2}+\frac{1}{3 !} x^{3}+\cdots .]$$

Determine two linearly independent solutions to the given differential equation on \((0, \infty)\) $$4 x^{2} y^{\prime \prime}+(1-4 x) y=0$$

(a) Verify that \(y(x)=x J_{0}(x)\) is a solution to the differential equation $$ y^{\prime \prime}+y=-J_{1}(x), \quad x>0 $$ (b) Use the result from (a) to establish that \(\int_{0}^{x} \cos (x-t) J_{0}(t) d t=x J_{0}(x), \quad x>0\) [Hint: Use Equation \((8.7 .15)\) and \((8.7 .16)\) to construct a particular solution to ( 11.6 .36 )

Consider the Chebyshev equation $$\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+a^{2} y=0, \quad (11.7.10)$$ where \(a\) is a constant. (a) Show that if \(a=N,\) a nonnegative integer, then Equation (11.7.10) has a polynomial solution of degree \(N .\) When suitably normalized, these polynomials are called the Chebyshev polynomials and are denoted by \(T_{N}(x).\) (b) Use Equation (11.7.10) to show that \(T_{N}(x)\) satisfies $$\left[\sqrt{1-x^{2}} T_{N}^{\prime}\right]^{\prime}+\frac{N}{\sqrt{1-x^{2}}} T_{N}=0.$$ (c) Use the result from (b) to prove that $$\int_{-1}^{1} \frac{T_{N}(x) T_{M}(x)}{\sqrt{1-x^{2}}} d x=0, \quad M \neq N.$$

Show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on \((0, \infty)\). $$x^{2} y^{\prime \prime}+x(1-x) y^{\prime}-(5+x) y=0$$

Determine terms up to and including \(x^{5}\) in two linearly independent power series solutions of the given differential equation. State the radius of convergence of the series solutions. $$y^{\prime \prime}+(\sin x) y^{\prime}+y=0.$$

If $$ \sum_{n=1}^{\infty}(n+1)(n+2) a_{n+1} x^{n}-\sum_{n=1}^{\infty} n a_{n-1} x^{n}=0 $$ show that for \(k=1,2,3, \ldots,\) we have $$ a_{2 k}=\frac{1 \cdot 3 \cdot 5 \cdots(2 k-1)}{(2 k+1) !} a_{0}, \quad a_{2 k+1}=\frac{2^{k+1} k !}{(2 k+2) !} a_{1} $$.

Show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on \((0, \infty)\). $$3 x^{2} y^{\prime \prime}+x(7+3 x) y^{\prime}+(1+6 x) y=0$$

Determine the Fourier-Bessel expansion in the functions \(J_{p}\left(\lambda_{n} x\right)\) for \(f(x)=x^{p},\) on the interval (0,1) [Here the \(\lambda_{n}\) denote the positive zeros of \(J_{p}(x) .\) Hint: You will need to use one of the results from Problem 11.]

Consider the differential equation $$x^{2} y^{\prime \prime}+x(1+2 N-x) y^{\prime}+N^{2} y=0, \quad \begin{array}{l}x>0, \quad (11.7.11)\end{array}$$ (a) Find the indicial equation, and show that it has only one root \(r=-N\) (b): If \(N\) is a nonnegative integer, show that the Frobenius series solution to Equation (11.7.11) terminates after \(N\) terms. (c) Determine the Frobenius series solutions when \(N=0,1,2,3.\) (d) Show that if \(N\) is a positive integer, then the Frobenius series solution to Equation (11.7.11) can be written as $$\begin{aligned} y(x) &=\\\ & x^{-N}\left[1+\sum_{k=1}^{N}(-1)^{k} \frac{N(N-1) \cdots(N+1-k)}{1^{2} \cdot 2^{2} \cdots k^{2}} x^{k}\right] \\ &=x^{-N}\left[1+\sum_{k=1}^{N}(-1)^{k} \prod_{i=1}^{k} \frac{(N+1-i)}{i^{2}} x^{k}\right]. \end{aligned}$$

Determine two linearly independent solutions to the given differential equation on \((0, \infty)\) $$x y^{\prime \prime}+y^{\prime}-2 y=0$$

Consider the differential equation $$x y^{\prime \prime}-(x-1) y^{\prime}-x y=0 \qquad (11.2 .18)$$ (a) Is \(x=0\) an ordinary point? (b) Determine the first three nonzero terms in each of two linearly independent series solutions to Equation \((11.2 .18)\) centered at \(x=1\). [Hint: Make the change of variables \(z=x-1\) and obtain a series solution in powers of \(z .]\) Give a lower bound on the radius of convergence of each of your solutions.