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# 3.2E: Exercises

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University
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In Exercises $$(3.2E.1)$$ to $$(3.2E.8)$$, find the power series in $$x$$ for the general solution.

## Exercise $$\PageIndex{1}$$

$$(1+x^2)y''+6xy'+6y=0$$

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## Exercise $$\PageIndex{2}$$

$$(1+x^2)y''+2xy'-2y=0$$

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## Exercise $$\PageIndex{3}$$

$$(1+x^2)y''-8xy'+20y=0$$

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## Exercise $$\PageIndex{4}$$

$$(1-x^2)y''-8xy'-12y=0$$

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## Exercise $$\PageIndex{5}$$

$$(1+2x^2)y''+7xy'+2y=0$$

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## Exercise $$\PageIndex{6}$$

$$\displaystyle{(1+x^2)y''+2xy'+{1\over4}y=0}$$

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## Exercise $$\PageIndex{7}$$

$$(1-x^2)y''-5xy'-4y=0$$

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## Exercise $$\PageIndex{8}$$

$$(1+x^2)y''-10xy'+28y=0$$

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## Exercise $$\PageIndex{9}$$

(a) Find the power series in $$x$$ for the general solution of $$y''+xy'+2y=0$$.

(b) For several choices of $$a_0$$ and $$a_1$$, use differential equations software to solve the initial value problem

\begin{equation}\label{eq:3.2E.1}
\end{equation}

numerically on $$(-5,5)$$.

(c) For fixed $$r$$ in $$\{1,2,3,4,5\}$$ graph

\begin{eqnarray*}
T_N(x)=\sum_{n=0}^N a_nx^n
\end{eqnarray*}

and the solution obtained in part (a) on $$(-r,r)$$. Continue increasing $$N$$ until there's no perceptible difference between the two graphs.

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## Exercise $$\PageIndex{10}$$

Follow the directions of Exercise $$(3.2E.9)$$ for the differential equation

\begin{eqnarray*}
y''+2xy'+3y=0.
\end{eqnarray*}

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In Exercises $$(3.2E.11)$$ to $$(3.2E.13)$$, find $$a_0$$, $$\dots$$, $$a_N$$ for $$N$$ at least $$7$$ in the power series solution $$y=\sum_{n=0}^\infty a_nx^n$$ of the initial value problem.

## Exercise $$\PageIndex{11}$$

$$(1+x^2)y''+xy'+y=0,\quad y(0)=2,\quad y'(0)=-1$$

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## Exercise $$\PageIndex{12}$$

$$(1+2x^2)y''-9xy'-6y=0,\quad y(0)=1,\quad y'(0)=-1$$

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## Exercise $$\PageIndex{13}$$

$$(1+8x^2)y''+2y=0,\quad y(0)=2,\quad y'(0)=-1$$

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## Exercise $$\PageIndex{14}$$

Do the next experiment for various choices of real numbers $$a_0$$, $$a_1$$, and $$r$$, with $$0<r<1/\sqrt2$$.

(a) Use differential equations software to solve the initial value problem

\begin{equation}\label{eq:3.2E.2}
\end{equation}

numerically on $$(-r,r)$$.

(b) For $$N=2$$, $$3$$, $$4$$, $$\dots$$, compute $$a_2$$, $$\dots$$, $$a_N$$ in the power series solution $$y=\sum_{n=0}^\infty a_nx^n$$ of \eqref{eq:3.2E.2}, and graph

\begin{eqnarray*}
T_N(x)=\sum_{n=0}^N a_nx^n
\end{eqnarray*}

and the solution obtained in part (a) on $$(-r,r)$$. Continue increasing $$N$$ until there's no perceptible difference between the two graphs.

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## Exercise $$\PageIndex{15}$$

Do part (a) and part (b) for several values of $$r$$ in $$(0,1)$$:

(a) Use differential equations software to solve the initial value problem

\begin{equation}\label{eq:3.2E.3}
\end{equation}

numerically on $$(-r,r)$$.

(b) For $$N=2$$, $$3$$, $$4$$, $$\dots$$, compute $$a_2$$, $$\dots$$, $$a_N$$ in the power series solution $$y=\sum_{n=0}^\infty a_nx^n$$ of \eqref{eq:3.2E.3}, and graph

\begin{eqnarray*}
T_N(x)=\sum_{n=0}^N a_nx^n
\end{eqnarray*}

and the solution obtained in part (a) on $$(-r,r)$$. Continue increasing $$N$$ until there's no perceptible difference between the two graphs. What happens to the required $$N$$ as $$r\to1$$?

(c) Try part (a) and part (b) with $$r=1.2$$. Explain your results.

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In Exercises $$(3.2E.16)$$ to $$(3.2E.20)$$, find the power series in $$-x_0$$ for the general solution.

## Exercise $$\PageIndex{16}$$

$$y''-y=0;\quad x_0=3$$

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## Exercise $$\PageIndex{17}$$

$$y''-(x-3)y'-y=0;\quad x_0=3$$

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## Exercise $$\PageIndex{18}$$

$$(1-4x+2x^2)y''+10(x-1)y'+6y=0;\quad x_0=1$$

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## Exercise $$\PageIndex{19}$$

$$(11-8x+2x^2)y''-16(x-2)y'+36y=0;\quad x_0=2$$

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## Exercise $$\PageIndex{20}$$

$$(5+6x+3x^2)y''+9(x+1)y'+3y=0;\quad x_0=-1$$

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In Exercises $$(3.2E.21)$$ to $$(3.2E.26)$$, find $$a_0$$, $$\dots$$, $$a_N$$ for $$N$$ at least $$7$$ in the power series $$y=\sum_{n=0}^\infty a_n(x-x_0)^n$$ for the solution of the initial value problem. Take $$x_0$$ to be the point where the initial conditions are imposed.

## Exercise $$\PageIndex{21}$$

$$(x^2-4)y''-xy'-3y=0,\quad y(0)=-1,\quad y'(0)=2$$

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## Exercise $$\PageIndex{22}$$

$$y''+(x-3)y'+3y=0,\quad y(3)=-2,\quad y'(3)=3$$

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## Exercise $$\PageIndex{23}$$

$$(5-6x+3x^2)y''+(x-1)y'+12y=0,\quad y(1)=-1,\quad y'(1)=1$$

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## Exercise $$\PageIndex{24}$$

$$(4x^2-24x+37)y''+y=0,\quad y(3)=4,\quad y'(3)=-6$$

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## Exercise $$\PageIndex{25}$$

$$(x^2-8x+14)y''-8(x-4)y'+20y=0,\quad y(4)=3,\quad y'(4)=-4$$

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## Exercise $$\PageIndex{26}$$

$$(2x^2+4x+5)y''-20(x+1)y'+60y=0,\quad y(-1)=3,\quad y'(-1)=-3$$

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## Exercise $$\PageIndex{27}$$

(a) Find a power series in $$x$$ for the general solution of

\begin{equation}\label{eq:3.2E.4}
(1+x^2)y''+4xy'+2y=0.
\end{equation}

(b) Use (a) and the formula

\begin{eqnarray*}
\end{eqnarray*}

for the sum of a geometric series to find a closed form expression for the general solution of \eqref{eq:3.2E.4} on $$(-1,1)$$.

(c) Show that the expression obtained in part (b) is actually the general solution of \eqref{eq:3.2E.4} on $$(-\infty,\infty)$$.

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## Exercise $$\PageIndex{28}$$

Use Theorem $$(3.2.2)$$ to show that the power series in $$x$$ for the general solution of

\begin{eqnarray*}
(1+\alpha x^2)y''+\beta xy'+\gamma y=0
\end{eqnarray*}

is

\begin{eqnarray*}
y=a_0\sum^\infty_{m=0}(-1)^m \left[\prod^{m-1}_{j=0}
p(2j)\right] {x^{2m}\over(2m)!} +
a_1\sum^\infty_{m=0}(-1)^m
\left[\prod^{m-1}_{j=0}p(2j+1)\right]
{x^{2m+1}\over(2m+1)!}.
\end{eqnarray*}

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## Exercise $$\PageIndex{29}$$

Use Exercise $$(3.2E.28)$$ to show that all solutions of

\begin{eqnarray*}
(1+\alpha x^2)y''+\beta xy'+\gamma y=0
\end{eqnarray*}

are polynomials if and only if

\begin{eqnarray*}
\alpha n(n-1)+\beta n+\gamma=\alpha(n-2r)(n-2s-1),
\end{eqnarray*}

where $$r$$ and $$s$$ are nonnegative integers.

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## Exercise $$\PageIndex{30}$$

(a) Use Exercise $$(3.2E.28)$$ to show that the power series in $$x$$ for the general solution of

\begin{eqnarray*}
(1-x^2)y''-2bxy'+\alpha(\alpha+2b-1)y=0
\end{eqnarray*}

is $$y=a_0y_1+a_1y_2$$, where

\begin{eqnarray*}
y_1&=&\sum_{m=0}^\infty \left[\prod_{j=0}^{m-1}(2j-\alpha)(2j+\alpha+2b-1) \right]{x^{2m}\over(2m)!}\\
\mbox{and}\\
y_2&=&\sum_{m=0}^\infty \left[\prod_{j=0}^{m-1}(2j+1-\alpha)(2j+\alpha+2b)\right]{x^{2m+1}\over(2m+1)!}.
\end{eqnarray*}

(b) Suppose $$2b$$ isn't a negative odd integer and $$k$$ is a nonnegative integer. Show that $$y_1$$ is a polynomial of degree $$2k$$ such that $$y_1(-x)=y_1(x)$$ if $$\alpha=2k$$, while $$y_2$$ is a polynomial of degree $$2k+1$$ such that $$y_2(-x)=-y_2(-x)$$ if $$\alpha=2k+1$$. Conclude that if $$n$$ is a nonnegative integer, then there's a polynomial $$P_n$$ of degree $$n$$ such that $$P_n(-x)=(-1)^nP_n(x)$$ and

\begin{equation}\label{eq:3.2E.5}
(1-x^2)P_n''-2bxP_n'+n(n+2b-1)P_n=0.
\end{equation}

(c) Show that \eqref{eq:3.2E.5} implies that

\begin{eqnarray*}
[(1-x^2)^b P_n']'=-n(n+2b-1)(1-x^2)^{b-1}P_n,
\end{eqnarray*}

and use this to show that if $$m$$ and $$n$$ are nonnegative integers, then

\begin{equation}\label{eq:3.2E.6}
\begin{array}{ll}
[(1-x^2)^bP_n']'P_m-[(1-x^2)^bP_m']'P_n=&\\
\left[m(m+2b-1)-n(n+2b-1)\right](1-x^2)^{b-1}P_mP_n.&
\end{array}
\end{equation}

(d) Now suppose $$b>0$$. Use \eqref{eq:3.2E.6} and integration by parts to show that if $$m\ne n$$, then

\begin{eqnarray*}
\int_{-1}^1 (1-x^2)^{b-1}P_m(x)P_n(x)\,dx=0.
\end{eqnarray*}

(We say that $$P_m$$ and $$P_n$$ are $$\textcolor{blue}{\mbox{orthogonal on}}$$ $$(-1,1)$$ $$\textcolor{blue}{\mbox{with respect to the weighting function}}$$ $$(1-x^2)^{b-1}$$.)
\end{alist}

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## Exercise $$\PageIndex{31}$$

(a) Use Exercise $$(3.2E.28)$$ to show that the power series in $$x$$ for the general solution of Hermite's equation

\begin{eqnarray*}
y''-2xy'+2\alpha y=0
\end{eqnarray*}

is $$y=a_0y_1+a_1y_1$$, where

\begin{eqnarray*}
y_1&=&\sum_{m=0}^\infty \left[\prod_{j=0}^{m-1}(2j-\alpha) \right]{2^mx^{2m}\over(2m)!}\\
\mbox{and}\\
y_2&=&\sum_{m=0}^\infty \left[\prod_{j=0}^{m-1}(2j+1-\alpha) \right]{2^mx^{2m+1}\over(2m+1)!}.
\end{eqnarray*}

(b) Suppose $$k$$ is a nonnegative integer. Show that $$y_1$$ is a polynomial of degree $$2k$$ such that $$y_1(-x)=y_1(x)$$ if $$\alpha=2k$$, while $$y_2$$ is a polynomial of degree $$2k+1$$ such that $$y_2(-x)=-y_2(-x)$$ if $$\alpha=2k+1$$. Conclude that if $$n$$ is a nonnegative integer then there's a polynomial $$P_n$$ of degree $$n$$ such that $$P_n(-x)=(-1)^nP_n(x)$$ and

\begin{equation}\label{eq:3.2E.7}
P_n''-2xP_n'+2nP_n=0.
\end{equation}

(c) Show that \eqref{eq:3.2E.7} implies that

\begin{eqnarray*}
[e^{-x^2}P_n']'=-2ne^{-x^2}P_n,
\end{eqnarray*}

and use this to show that if $$m$$ and $$n$$ are nonnegative integers, then

\begin{equation}\label{eq:3.2E.8}
[e^{-x^2}P_n']'P_m-[e^{-x^2}P_m']'P_n= 2(m-n)e^{-x^2}P_mP_n.
\end{equation}

(d) Use \eqref{eq:3.2E.8} and integration by parts to show that if $$m\ne n$$, then

\begin{eqnarray*}
\int_{-\infty}^\infty e^{-x^2}P_m(x)P_n(x)\,dx=0.
\end{eqnarray*}

(We say that $$P_m$$ and $$P_n$$ are $$\textcolor{blue}{\mbox{orthogonal on}}$$ $$(-\infty,\infty)$$ $$\textcolor{blue}{\mbox{with respect to the weighting function}}$$ $$e^{-x^2}$$.)

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## Exercise $$\PageIndex{32}$$

Consider the equation

\begin{equation}\label{eq:3.2E.9}
\left(1+\alpha x^3\right)y''+\beta x^2y'+\gamma xy=0,
\end{equation}

and let $$p(n)=\alpha n(n-1)+\beta n+\gamma$$. (The special case $$y''-xy=0$$ of \eqref{eq:3.2E.9} is Airy's equation.)

(a) Modify the argument used to prove Theorem $$(3.2.2)$$ to show that

\begin{eqnarray*}
y=\sum_{n=0}^\infty a_nx^n
\end{eqnarray*}

is a solution of \eqref{eq:3.2E.9} if and only if $$a_2=0$$ and

\begin{eqnarray*}
\end{eqnarray*}

(b) Show from (a) that $$a_n=0$$ unless $$n=3m$$ or $$n=3m+1$$ for some nonnegative integer $$m$$, and that

\begin{eqnarray*}
\mbox{and}\\
\end{eqnarray*}

where $$a_0$$ and $$a_1$$ may be specified arbitrarily.

(c) Conclude from (b) that the power series in $$x$$ for the general solution of \eqref{eq:3.2E.9} is

\begin{eqnarray*}
\begin{array}{l}
y=\displaystyle{a_0\sum^\infty_{m=0}(-1)^m \left[\prod^{m-1}_{j=0} {p(3j)\over3j+2}\right] {x^{3m}\over3^m m!}}\\
\end{array}
\end{eqnarray*}

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In Exercises $$(3.2E.33)$$ to $$(3.2E.37)$$, use the method of Exercise $$(3.2E.32)$$ to find the power series in $$x$$ for the general solution.

## Exercise $$\PageIndex{33}$$

$$y''-xy=0$$

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## Exercise $$\PageIndex{34}$$

$$(1-2x^3)y''-10x^2y'-8xy=0$$

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## Exercise $$\PageIndex{35}$$

$$(1+x^3)y''+7x^2y'+9xy=0$$

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## Exercise $$\PageIndex{36}$$

$$(1-2x^3)y''+6x^2y'+24xy=0$$

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## Exercise $$\PageIndex{37}$$

$$(1-x^3)y''+15x^2y'-63xy=0$$

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## Exercise $$\PageIndex{38}$$

Consider the equation

\begin{equation}\label{eq:3.2E.10}
\left(1+\alpha x^{k+2}\right)y''+\beta x^{k+1}y'+\gamma x^ky=0,
\end{equation}

where $$k$$ is a positive integer, and let $$p(n)=\alpha n(n-1)+\beta n+\gamma$$.

(a) Modify the argument used to prove Theorem $$(3.2.2)$$ to show that

\begin{eqnarray*}
y=\sum_{n=0}^\infty a_nx^n
\end{eqnarray*}

is a solution of \eqref{eq:3.2E.10} if and only if $$a_n=0$$ for $$2\le n\le k+1$$ and

\begin{eqnarray*}
\end{eqnarray*}

(b) Show from (a) that $$a_n=0$$ unless $$n=(k+2)m$$ or $$n=(k+2)m+1$$ for some nonnegative integer $$m$$, and that

\begin{eqnarray*}
\mbox{and}\\
\end{eqnarray*}

where $$a_0$$ and $$a_1$$ may be specified arbitrarily.

(c) Conclude from (b) that the power series in $$x$$ for the general solution of \eqref{eq:3.2E.10} is

\begin{eqnarray*}
\begin{array}{l}
y=a_0\displaystyle{\sum^\infty_{m=0}(-1)^m \left[\prod^{m-1}_{j=0} {p\left((k+2)j\right)\over(k+2)(j+1)-1}\right] {x^{(k+2)m}\over(k+2)^m m!}}\\
\end{array}
\end{eqnarray*}

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In Exercises $$(3.2E.39)$$ to $$(3.2E.44)$$, use the method of Exercise $$(3.2E.38)$$ to find the power series in $$x$$ for the general solution.

## Exercise $$\PageIndex{39}$$

$$(1+2x^5)y''+14x^4y'+10x^3y=0$$

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## Exercise $$\PageIndex{40}$$

$$y''+x^2y=0$$

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## Exercise $$\PageIndex{41}$$

$$y''+x^6y'+7x^5y=0$$

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## Exercise $$\PageIndex{42}$$

$$(1+x^8)y''-16x^7y'+72x^6y=0$$

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## Exercise $$\PageIndex{43}$$

$$(1-x^6)y''-12x^5y'-30x^4y=0$$

## Exercise $$\PageIndex{44}$$
$$y''+x^5y'+6x^4y=0$$