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3.3E: Exercises

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In Exercises $$(3.3E.1)$$ to $$(3.3E.12)$$, find the coefficients $$a_0$$, $$\dots$$, $$a_N$$ for $$N$$ at least $$7$$ in the series solution $$y=\sum_{n=0}^\infty a_nx^n$$ of the initial value problem.

Exercise $$\PageIndex{1}$$

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

Answer

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Exercise $$\PageIndex{2}$$

$$(1+x+2x^2)y''+(2+8x)y'+4y=0,\quad y(0)=-1,\quad y'(0)=2$$

Answer

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Exercise $$\PageIndex{3}$$

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

Answer

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Exercise $$\PageIndex{4}$$

$$(1+x+3x^2)y''+(2+15x)y'+12y=0,\quad y(0)=0,\quad y'(0)=1$$

Answer

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Exercise $$\PageIndex{5}$$

$$(2+x)y''+(1+x)y'+3y=0,\quad y(0)=4,\quad y'(0)=3$$

Answer

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Exercise $$\PageIndex{6}$$

$$(3+3x+x^2)y''+(6+4x)y'+2y=0,\quad y(0)=7,\quad y'(0)=3$$

Answer

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Exercise $$\PageIndex{7}$$

$$(4+x)y''+(2+x)y'+2y=0,\quad y(0)=2,\quad y'(0)=5$$

Answer

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Exercise $$\PageIndex{8}$$

$$(2-3x+2x^2)y''-(4-6x)y'+2y=0,\quad y(1)=1,\quad y'(1)=-1$$

Answer

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Exercise $$\PageIndex{9}$$

$$(3x+2x^2)y''+10(1+x)y'+8y=0,\quad y(-1)=1,\quad y'(-1)=-1$$

Answer

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Exercise $$\PageIndex{10}$$

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

Answer

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Exercise $$\PageIndex{11}$$

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

Answer

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Exercise $$\PageIndex{12}$$

$$x^2y''-(6-7x)y'+8y=0,\quad y(1)=1,\quad y'(1)=-2$$

Answer

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Exercise $$\PageIndex{13}$$

Do the following 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

\label{eq:3.3E.1}
(1+x+2x^2)y''+(1+7x)y'+2y=0,\quad y(0)=a_0,\quad y'(0)=a_1,

numerically on $$(-r,r)$$. (See Example $$(3.3.1)$$.)

(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:3.3E.1}, and graph

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

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

Answer

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Exercise $$\PageIndex{14}$$

Do the following experiment for various choices of real numbers $$a_0$$, $$a_1$$, and $$r$$, with $$0<r<2$$.

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

\label{eq:3.3E.2}
(3+x)y''+(1+2x)y'-(2-x)y=0,\quad y(-1)=a_0,\quad y'(-1)=a_1,

numerically on $$(-1-r,-1+r)$$. (See Example $$(3.3.2)$$. Why this interval?)

(b) For $$N=2$$, $$3$$, $$4$$, $$\dots$$, compute $$a_2,\dots,a_N$$ in the power series solution

\begin{eqnarray*}
y=\sum_{n=0}^\infty a_n(x+1)^n
\end{eqnarray*}

of \eqref{eq:3.3E.2}, and graph

\begin{eqnarray*}
T_N(x)=\sum_{n=0}^N a_n(x+1)^n
\end{eqnarray*}

and the solution obtained in (a) on $$(-1-r,-1+r)$$. Continue increasing $$N$$ until there's no perceptible difference between the two graphs.

Answer

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Exercise $$\PageIndex{15}$$

Do the following experiment for several choices of $$a_0$$, $$a_1$$, and $$r$$, with $$r>0$$.

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

\label{eq:3.3E.3}
y''+3xy'+(4+2x^2)y=0,\quad y(0)=a_0,\quad y'(0)=a_1,

numerically on $$(-r,r)$$. (See Example $$(3.3.3)$$.)

(b) Find the coefficients $$a_0$$, $$a_1$$, $$\dots$$, $$a_N$$ in the power series solution $$y=\sum_{n=0}^\infty a_nx^n$$ of \eqref{eq:3.3E.3}, and graph

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

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

Answer

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Exercise $$\PageIndex{16}$$

Do the following experiment for several choices of $$a_0$$ and $$a_1$$.

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

\label{eq:3.3E.4}
(1-x)y''-(2-x)y'+y=0,\quad y(0)=a_0,\quad y'(0)=a_1,

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

(b) Find the coefficients $$a_0$$, $$a_1$$, $$\dots$$, $$a_N$$ in the power series solution $$y=\sum_{n=0}^Na_nx^n$$ of \eqref{eq:3.3E.4}, and graph

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

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

Answer

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Exercise $$\PageIndex{17}$$

Follow the directions of Exercise $$(3.3E.16)$$ for the initial value problem

\begin{eqnarray*}
(1+x)y''+3y'+32y=0,\quad y(0)=a_0,\quad y'(0)=a_1.
\end{eqnarray*}

Answer

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Exercise $$\PageIndex{18}$$

Follow the directions of Exercise $$(3.3E.16)$$ for the initial value problem

\begin{eqnarray*}
(1+x^2)y''+y'+2y=0,\quad y(0)=a_0,\quad y'(0)=a_1.
\end{eqnarray*}

Answer

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In Exercises $$(3.3E.19)$$ to $$(3.3E.28)$$, find the coefficients $$a_0$$, $$\dots$$, $$a_N$$ for $$N$$ at least $$7$$ in the series solution

\begin{eqnarray*}
y=\sum_{n=0}^\infty a_n(x-x_0)^n
\end{eqnarray*}

of the initial value problem. Take $$x_0$$ to be the point where the initial conditions are imposed.

Exercise $$\PageIndex{19}$$

$$(2+4x)y''-4y'-(6+4x)y=0,\quad y(0)=2,\quad y'(0)=-7$$

Answer

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Exercise $$\PageIndex{20}$$

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

Answer

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Exercise $$\PageIndex{21}$$

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

Answer

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Exercise $$\PageIndex{22}$$

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

Answer

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Exercise $$\PageIndex{23}$$

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

Answer

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Exercise $$\PageIndex{24}$$

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

Answer

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Exercise $$\PageIndex{25}$$

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

Answer

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Exercise $$\PageIndex{26}$$

$$(10-2x)y''+(1+x)y=0,\quad y(2)=2,\quad y'(2)=-4$$

Answer

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Exercise $$\PageIndex{27}$$

$$(7+x)y''+(8+2x)y'+(5+x)y=0,\quad y(-4)=1,\quad y'(-4)=2$$

Answer

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Exercise $$\PageIndex{28}$$

$$(6+4x)y''+(1+2x)y=0,\quad y(-1)=-1,\quad y'(-1)=2$$

Answer

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Exercise $$\PageIndex{29}$$

Show that the coefficients in the power series in $$x$$ for the general solution of

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

satisfy the recurrence relation

\begin{eqnarray*}
a_{n+2}=-{\gamma+\alpha n\over n+2}\,a_{n+1}-{\beta n(n-1)+\delta n+\epsilon\over(n+2)(n+1)}\, a_n.
\end{eqnarray*}

Answer

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Exercise $$\PageIndex{30}$$

(a) Let $$\alpha$$ and $$\beta$$ be constants, with $$\beta\ne0$$. Show that $$y=\sum_{n=0}^\infty a_nx^n$$ is a solution of

\label{eq:3.3E.5}
(1+\alpha x+\beta x^2)y''+(2\alpha+4\beta x)y'+2\beta y=0

if and only if

\label{eq:3.3E.6}
a_{n+2}+\alpha a_{n+1}+\beta a_n=0,\quad n\ge0.

An equation of this form is called a $$\textcolor{blue}{\mbox{second order homogeneous linear difference equation}}$$. The polynomial $$p(r)=r^2+\alpha r+\beta$$ is called the $$\textcolor{blue}{\mbox{characteristic polynomial}}$$ of \eqref{eq:3.3E.6}. If $$r_1$$ and $$r_2$$ are the zeros of $$p$$, then $$1/r_1$$ and $$1/r_2$$ are the zeros of

\begin{eqnarray*}
P_0(x)=1+\alpha x+\beta x^2.
\end{eqnarray*}

(b) Suppose $$p(r)=(r-r_1)(r-r_2)$$ where $$r_1$$ and $$r_2$$ are real and distinct, and let $$\rho$$ be the smaller of the two numbers $$\{1/|r_1|,1/|r_2|\}$$. Show that if $$c_1$$ and $$c_2$$ are constants then the sequence

\begin{eqnarray*}
a_n=c_1r_1^n+c_2r_2^n,\quad n\ge0
\end{eqnarray*}

satisfies \eqref{eq:3.3E.6}. Conclude from this that any function of the form

\begin{eqnarray*}
y=\sum_{n=0}^\infty (c_1r_1^n+c_2r_2^n)x^n
\end{eqnarray*}

is a solution of \eqref{eq:3.3E.5} on $$(-\rho,\rho)$$.

(c) Use (b) and the formula for the sum of a geometric series to show that the functions

\begin{eqnarray*}
y_1={1\over1-r_1x}\quad\mbox{ and }\quad y_2={1\over1-r_2x}
\end{eqnarray*}

form a fundamental set of solutions of \eqref{eq:3.3E.5} on $$(-\rho,\rho)$$.

(d) Show that $$\{y_1,y_2\}$$ is a fundamental set of solutions of \eqref{eq:3.3E.5} on any interval that doesn't contain either $$1/r_1$$ or $$1/r_2$$.

(e) Suppose $$p(r)=(r-r_1)^2$$, and let $$\rho=1/|r_1|$$. Show that if $$c_1$$ and $$c_2$$ are constants then the sequence

\begin{eqnarray*}
a_n=(c_1+c_2n)r_1^n,\quad n\ge0
\end{eqnarray*}

satisfies \eqref{eq:3.3E.6}. Conclude from this that any function of the form

\begin{eqnarray*}
y=\sum_{n=0}^\infty (c_1+c_2n)r_1^nx^n
\end{eqnarray*}

is a solution of \eqref{eq:3.3E.5} on $$(-\rho,\rho)$$.

(f) Use (e) and the formula for the sum of a geometric series to show that the functions

\begin{eqnarray*}
y_1={1\over1-r_1x}\quad\mbox{ and }\quad y_2={x\over(1-r_1x)^2}
\end{eqnarray*}

form a fundamental set of solutions of \eqref{eq:3.3E.5} on $$(-\rho,\rho)$$.

(g) Show that $$\{y_1,y_2\}$$ is a fundamental set of solutions of \eqref{eq:3.3E.5} on any interval that does not contain $$1/r_1$$.

Answer

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Exercise $$\PageIndex{31}$$

Use the results of Exercise $$(3.3E.30)$$ to find the general solution of the given equation on any interval on which polynomial multiplying $$y''$$ has no zeros.

(a) $$(1+3x+2x^2)y''+(6+8x)y'+4y=0$$

(b) $$(1-5x+6x^2)y''-(10-24x)y'+12y=0$$

(c) $$(1-4x+4x^2)y''-(8-16x)y'+8y=0$$

(d) $$(4+4x+x^2)y''+(8+4x)y'+2y=0$$

(e) $$(4+8x+3x^2)y''+(16+12x)y'+6y=0$$

Answer

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In Exercises $$(3.3E.32)$$ to $$(3.3E.38)$$, find the coefficients $$a_0$$, $$\dots$$, $$a_N$$ for $$N$$ at least $$7$$ in the series solution $$y=\sum_{n=0}^\infty a_nx^n$$ of the initial value problem.

Exercise $$\PageIndex{32}$$

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

Answer

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Exercise $$\PageIndex{33}$$

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

Answer

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Exercise $$\PageIndex{34}$$

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

Answer

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Exercise $$\PageIndex{35}$$

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

Answer

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Exercise $$\PageIndex{36}$$

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

Answer

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Exercise $$\PageIndex{37}$$

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

Answer

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Exercise $$\PageIndex{38}$$

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

Answer

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Exercise $$\PageIndex{39}$$

Find power series in $$x$$ for the solutions $$y_1$$ and $$y_2$$ of

\begin{eqnarray*}
y''+4xy'+(2+4x^2)y=0
\end{eqnarray*}

such that $$y_1(0)=1$$, $$y'_1(0)=0$$, $$y_2(0)=0$$, $$y'_2(0)=1$$, and identify $$y_1$$ and $$y_2$$ in terms of familiar elementary functions.

Answer

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In Exercises $$(3.3E.40)$$ tp $$(3.3E.49)$$, find the coefficients $$a_0$$, $$\dots$$, $$a_N$$ for $$N$$ at least $$7$$ in the series solution

\begin{eqnarray*}
y=\sum_{n=0}^\infty a_n(x-x_0)^n
\end{eqnarray*}

of the initial value problem. Take $$x_0$$ to be the point where the initial conditions are imposed.

Exercise $$\PageIndex{40}$$

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

Answer

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Exercise $$\PageIndex{41}$$

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

Answer

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Exercise $$\PageIndex{42}$$

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

Answer

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Exercise $$\PageIndex{43}$$

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

Answer

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Exercise $$\PageIndex{44}$$

$$y''+(13+12x+3x^2)y'+(5+2x),\quad y(-2)=2,\quad y'(-2)=-3$$

Answer

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Exercise $$\PageIndex{45}$$

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

Answer

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Exercise $$\PageIndex{46}$$

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

Answer

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Exercise $$\PageIndex{47}$$

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

Answer

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Exercise $$\PageIndex{48}$$

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

Answer

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Exercise $$\PageIndex{49}$$

$$(16-11x+2x^2)y''+(10-6x+x^2)y'-(2-x)y,\quad y(3)=1,\quad y'(3)=-2$$

Answer

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