2.1E: Exercises

Exercise $\PageIndex{1}$

(a) Verify that $y_1=e^{2x}$ and $y_2=e^{5x}$ are solutions of

$$\begin{equation}\label{eq:2.1.1} y''-7y'+10y=0 \end{equation}$$

on $(-\infty,\infty)$.

(b) Verify that if $c_1$ and $c_2$ are arbitrary constants then $y=c_1e^{2x}+c_2e^{5x}$ is a solution of $\eqref{eq:2.1.1}$ on $(-\infty,\infty)$.

(c) Solve the initial value problem

$$\begin{eqnarray*} y''-7y'+10y=0,\quad y(0)=-1,\quad y'(0)=1. \end{eqnarray*}$$

(d) Solve the initial value problem

$$\begin{eqnarray*} y''-7y'+10y=0,\quad y(0)=k_0,\quad y'(0)=k_1. \end{eqnarray*}$$

Answer

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Exercise $\PageIndex{2}$

(a) Verify that $y_1=e^x\cos x$ and $y_2=e^x\sin x$ are solutions of

$$\begin{equation}\label{eq:2.1.2} y''-2y'+2y=0 \end{equation}$$

on $(-\infty,\infty)$.

(b) Verify that if $c_1$ and $c_2$ are arbitrary constants then $y=c_1e^x\cos x+c_2e^x\sin x$ is a solution of $\eqref{eq:2.1.2}$ on $(-\infty,\infty)$.

(c) Solve the initial value problem

$$\begin{eqnarray*} y''-2y'+2y=0,\quad y(0)=3,\quad y'(0)=-2. \end{eqnarray*}$$

(d) Solve the initial value problem

$$\begin{eqnarray*} y''-2y'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1. \end{eqnarray*}$$

Answer

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Exercise $\PageIndex{3}$

(a) Verify that $y_1=e^x$ and $y_2=xe^x$ are solutions of

$$\begin{equation}\label{eq:2.1.3} y''-2y'+y=0 \end{equation}$$

on $(-\infty,\infty)$.

(b) Verify that if $c_1$ and $c_2$ are arbitrary constants then $y=e^x(c_1+c_2x)$ is a solution of $\eqref{eq:2.1.3}$ on $(-\infty,\infty)$.

(c) Solve the initial value problem

$$\begin{eqnarray*} y''-2y'+y=0,\quad y(0)=7,\quad y'(0)=4. \end{eqnarray*}$$

(d) Solve the initial value problem

$$\begin{eqnarray*} y''-2y'+y=0,\quad y(0)=k_0,\quad y'(0)=k_1. \end{eqnarray*}$$

Answer

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Exercise $\PageIndex{4}$

(a) Verify that $y_1=1/(x-1)$ and $y_2=1/(x+1)$ are solutions of

$$\begin{equation}\label{eq:2.1.4} (x^2-1)y''+4xy'+2y=0 \end{equation}$$

on $(-\infty,-1)$, $(-1,1)$, and $(1,\infty)$. What is the general solution of $\eqref{eq:2.1.4}$ on each of these intervals?

(b) Solve the initial value problem

$$\begin{eqnarray*} (x^2-1)y''+4xy'+2y=0,\quad y(0)=-5,\quad y'(0)=1. \end{eqnarray*}$$

What is the interval of validity of the solution?

(c) Graph the solution of the initial value problem.

(d) Verify Abel's formula for $y_1$ and $y_2$, with $x_0=0$.

Answer

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Exercise $\PageIndex{5}$

Compute the Wronskians of the given sets of functions.

(a) $\{1, e^x\}$

(b)$\{e^x, e^x \sin x\}$

(c) $\{x+1, x^2+2\}$

(d) $\{ x^{1/2}, x^{-1/3}\}$

(e) $\left\{\displaystyle \frac{\sin x}{x}, \frac{\cos x}{x}\right\}$

(f) $\{ x \ln|x|, x^2\ln|x|\}$

(g) $\{e^x\cos\sqrt x, e^x\sin\sqrt x\}$

Answer

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Exercise $\PageIndex{6}$

Find the Wronskian of a given set $\{y_1,y_2\}$ of solutions of

$$\begin{eqnarray*} y''+3(x^2+1)y'-2y=0, \end{eqnarray*}$$

given that $W(\pi)=0$.

Answer

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Exercise $\PageIndex{7}$

Find the Wronskian of a given set $\{y_1,y_2\}$ of solutions of

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

given that $W(0)=1$. (This is Legendre's equation.)

Answer

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Exercise $\PageIndex{8}$

Find the Wronskian of a given set $\{y_1,y_2\}$ of solutions of

$$\begin{eqnarray*} x^2y''+xy'+(x^2-\nu^2)y=0 , \end{eqnarray*}$$

given that $W(1)=1$. (This is Bessel's equation.)

Answer

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Exercise $\PageIndex{9}$

(This exercise shows that if you know one nontrivial solution of $y''+p(x)y'+q(x)y=0$, you can use Abel's formula to find another.)

Suppose $p$ and $q$ are continuous and $y_1$ is a solution of

$$\begin{equation}\label{eq:2.1.5} y''+p(x)y'+q(x)y=0 \end{equation}$$

that has no zeros on $(a,b)$. Let $P(x)=\int p(x)\,dx$ be any antiderivative of $p$ on \)(a,b)\).

(a) Show that if $K$ is an arbitrary nonzero constant and $y_2$ satisfies

$$\begin{equation}\label{eq:2.1.6} y_1y_2'-y_1'y_2=Ke^{-P(x)} \end{equation}$$

on $(a,b)$, then $y_2$ also satisfies $\eqref{eq:2.1.5}$ on $(a,b)$, and $\{y_1,y_2\}$ is a fundmental set of solutions on $\eqref{eq:2.1.5}$ on $(a,b)$.

(b) Conclude from (a) that if $y_2=uy_1$ where $u'=K\displaystyle{e^{-P(x)}\over y_1^2(x)}$, then $\{y_1,y_2\}$ is a fundamental set of solutions of $\eqref{eq:2.1.5}$ on $(a,b)$.

Answer

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In Exercises $(2.1E.10)$ to $(2.1E.23)$ use the method suggested by Exercise $(2.1E.9)$ to find a second solution $y_2$ that isn't a constant multiple of the solution $y_1$. Choose $K$ conveniently to simplify $y_2$.

Exercise $\PageIndex{10}$

$y''-2y'-3y=0; \quad y_1=e^{3x}$

Answer

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Exercise $\PageIndex{11}$

$y''-6y'+9y=0; \quad y_1=e^{3x}$

Answer

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Exercise $\PageIndex{12}$

$y''-2ay'+a^2y=0; \mbox{(\(a=\) constant)}; \quad y_1=e^{ax}$

Answer

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Exercise $\PageIndex{13}$

$x^2y''+xy'-y=0; \quad y_1=x$

Answer

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Exercise $\PageIndex{14}$

$x^2y''-xy'+y=0; \quad y_1=x$

Answer

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Exercise $\PageIndex{15}$

$x^2y''-(2a-1)xy'+a^2y=0; \mbox{(\(a=\) nonzero constant), \(x>0\);} \quad y_1=x^a$

Answer

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Exercise $\PageIndex{16}$

$4x^2y''-4xy'+(3-16x^2)y=0; \quad y_1=x^{1/2}e^{2x}$

Answer

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Exercise $\PageIndex{17}$

$(x-1)y''-xy'+y=0; \quad y_1=e^x$

Answer

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Exercise $\PageIndex{18}$

$x^2y''-2xy'+(x^2+2)y=0; \quad y_1=x\cos x$

Answer

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Exercise $\PageIndex{19}$

$4x^2(\sin x)y''-4x(x\cos x+\sin x)y'+(2x\cos x+3\sin x)y=0; \quad y_1=x^{1/2}$

Answer

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Exercise $\PageIndex{20}$

$(3x-1)y''-(3x+2)y'-(6x-8)y=0; \quad y_1=e^{2x}$

Answer

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Exercise $\PageIndex{21}$

$(x^2-4)y''+4xy'+2y=0; \quad y_1=\displaystyle{1\over x-2}$

Answer

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Exercise $\PageIndex{22}$

$(2x+1)xy''-2(2x^2-1)y'-4(x+1)y=0;\quad y_1=\displaystyle{1\over x}$

Answer

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Exercise $\PageIndex{23}$

$(x^2-2x)y''+(2-x^2)y'+(2x-2)y=0;\quad y_1=e^x$

Answer

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Exercise $\PageIndex{24}$

Suppose $p$ and $q$ are continuous on an open interval $(a,b)$ and let $x_0$ be in $(a,b)$. Use Theorem $(2.1.1)$ to show that the only solution of the initial value problem

$$\begin{eqnarray*} y''+p(x)y'+q(x)y=0,\quad y(x_0)=0,\quad y'(x_0)=0 \end{eqnarray*}$$

on $(a,b)$ is the trivial solution $y\equiv0$.

Answer

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Exercise $\PageIndex{25}$

Suppose $P_0$, $P_1$, and $P_2$ are continuous on $(a,b)$ and let $x_0$ be in $(a,b)$. Show that if either of the following statements is true then $P_0(x)=0$ for some $x$ in $(a,b)$.

(a) The initial value problem

$$\begin{eqnarray*} P_0(x)y''+P_1(x)y'+P_2(x)y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1 \end{eqnarray*}$$

has more than one solution on $(a,b)$.

(b) The initial value problem

$$\begin{eqnarray*} P_0(x)y''+P_1(x)y'+P_2(x)y=0,\quad y(x_0)=0,\quad y'(x_0)=0 \end{eqnarray*}$$

has a nontrivial solution on $(a,b)$.

Answer

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Exercise $\PageIndex{26}$

Suppose $p$ and $q$ are continuous on $(a,b)$ and $y_1$ and $y_2$ are solutions of

$$\begin{equation}\label{eq:2.1.7} y''+p(x)y'+q(x)y=0 \end{equation}$$

on $(a,b)$. Let

$$\begin{eqnarray*} z_1=\alpha y_1+\beta y_2\quad \mbox{and} \quad z_2=\gamma y_1+\delta y_2, \end{eqnarray*}$$

where $\alpha$, $\beta$, $\gamma$, and $\delta$ are constants. Show that if $\{z_1,z_2\}$ is a fundamental set of solutions of $\eqref{eq:2.1.7}$ on $(a,b)$ then so is $\{y_1,y_2\}$.

Answer

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Exercise $\PageIndex{27}$

Suppose $p$ and $q$ are continuous on $(a,b)$ and $\{y_1,y_2\}$ is a fundamental set of solutions of

$$\begin{equation}\label{eq:2.1.8} y''+p(x)y'+q(x)y=0 \end{equation}$$

on $(a,b)$. Let

$$\begin{eqnarray*} z_1=\alpha y_1+\beta y_2\quad \mbox{and} \quad z_2=\gamma y_1+\delta y_2, \end{eqnarray*}$$

where $\alpha,\beta,\gamma$, and $\delta$ are constants. Show that $\{z_1,z_2\}$ is a fundamental set of solutions of $\eqref{eq:2.1.8}$ on $(a,b)$ if and only if $\alpha\gamma-\beta\delta\ne0$.

Answer

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Exercise $\PageIndex{28}$

Suppose $y_1$ is differentiable on an interval $(a,b)$ and $y_2=ky_1$, where $k$ is a constant. Show that the Wronskian of $\{y_1,y_2\}$ is identically zero on $(a,b)$.

Answer

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Exercise $\PageIndex{29}$

Let

$$\begin{eqnarray*} y_1=x^3 \quad \mbox{ and } \quad y_2=\left\{\begin{array}{rl} x^3,&x\ge 0,\\ -x^3,&x<0.\end{array}\right. \end{eqnarray*}$$

(a) Show that the Wronskian of $\{y_1,y_2\}$ is defined and identically zero on $(-\infty,\infty)$.

(b) Suppose $a<0<b$. Show that $\{y_1,y_2\}$ is linearly independent on $(a,b)$.

(c) Use Exercise $(2.1E.25)$ part (b) to show that these results don't contradict Theorem $(2.1.5)$, because neither $y_1$ nor $y_2$ can be a solution of an equation

$$\begin{eqnarray*} y''+p(x)y'+q(x)y=0 \end{eqnarray*}$$

on $(a,b)$ if $p$ and $q$ are continuous on $(a,b)$.

Answer

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Exercise $\PageIndex{30}$

Suppose $p$ and $q$ are continuous on $(a,b)$ and $\{y_1,y_2\}$ is a set of solutions of

$$\begin{eqnarray*} y''+p(x)y'+q(x)y=0 \end{eqnarray*}$$

on $(a,b)$ such that either $y_1(x_0)=y_2(x_0)=0$ or $y_1'(x_0)=y_2'(x_0)=0$ for some $x_0$ in $(a,b)$. Show that $\{y_1,y_2\}$ is linearly dependent on $(a,b)$.

Answer

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Exercise $\PageIndex{31}$

Suppose $p$ and $q$ are continuous on $(a,b)$ and $\{y_1,y_2\}$ is a fundamental set of solutions of

$$\begin{eqnarray*} y''+p(x)y'+q(x)y=0 \end{eqnarray*}$$

on $(a,b)$. Show that if $y_1(x_1)=y_1(x_2)=0$, where $a<x_1<x_2<b$, then $y_2(x)=0$ for some $x$ in $(x_1,x_2)$.

Hint: Show that if $y_2$ has no zeros in $(x_1,x_2)$, then $y_1/y_2$ is either strictly increasing or strictly decreasing on $(x_1,x_2)$, and deduce a contradiction.

Answer

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Exercise $\PageIndex{32}$

Suppose $p$ and $q$ are continuous on $(a,b)$ and every solution of

$$\begin{equation}\label{eq:2.1.9} y''+p(x)y'+q(x)y=0 \end{equation}$$

on $(a,b)$ can be written as a linear combination of the twice differentiable functions $\{y_1,y_2\}$. Use Theorem $(2.1.1)$ to show that $y_1$ and $y_2$ are themselves solutions of $\eqref{eq:2.1.9}$ on $(a,b)$.

Answer

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Exercise $\PageIndex{33}$

Suppose $p_1$, $p_2$, $q_1$, and $q_2$ are continuous on $(a,b)$ and the equations

$$\begin{eqnarray*} y''+p_1(x)y'+q_1(x)y=0 \quad \mbox{and} \quad y''+p_2(x)y'+q_2(x)y=0 \end{eqnarray*}$$

have the same solutions on $(a,b)$. Show that $p_1=p_2$ and $q_1=q_2$ on $(a,b)$.

Hint: Use Abel's formula.

Answer

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Exercise $\PageIndex{34}$

(For this exercise you have to know about $3\times 3$ determinants.)

Show that if $y_1$ and $y_2$ are twice continuously differentiable on $(a,b)$ and the Wronskian $W$ of $\{y_1,y_2\}$ has no zeros in $(a,b)$ then the equation

$$\begin{eqnarray*} \frac{1}{W} \left| \begin{array}{ccc} y & y_1 & y_2 \\ y' & y'_1 & y'_2 \\ y'' & y_1'' & y_2'' \end{array} \right|=0 \end{eqnarray*}$$

can be written as

$$\begin{equation}\label{eq:2.1.10} y''+p(x)y'+q(x)y=0, \end{equation}$$

where $p$ and $q$ are continuous on $(a,b)$ and $\{y_1,y_2\}$ is a fundamental set of solutions of $\eqref{eq:2.1.10}$ on $(a,b)$.

Hint: Expand the determinant by cofactors of its first column.

Answer

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Exercise $\PageIndex{35}$

Use the method suggested by Exercise $(2.1E.34)$ to find a linear homogeneous equation for which the given functions form a fundamental set of solutions on some interval.

(a) $e^x \cos 2x, \quad e^x \sin 2x$

(b) $x, \quad e^{2x}$

(c) $x, \quad x \ln x$

(d) $\cos (\ln x), \quad \sin (\ln x)$

(e) $\cosh x, \quad \sinh x$

(f) $x^2-1, \quad x^2+1$

Answer

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Exercise $\PageIndex{36}$

Suppose $p$ and $q$ are continuous on $(a,b)$ and $\{y_1,y_2\}$ is a fundamental set of solutions of

$$\begin{equation}\label{eq:2.1E.11} y''+p(x)y'+q(x)y=0 \end{equation}$$

on $(a,b)$. Show that if $y$ is a solution of $\eqref{eq:2.1E.11}$ on $(a,b)$, there's exactly one way to choose $c_1$ and $c_2$ so that $y=c_1y_1+c_2y_2$ on $(a,b)$.

Answer

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Exercise $\PageIndex{37}$

Suppose $p$ and $q$ are continuous on $(a,b)$ and $x_0$ is in $(a,b)$. Let $y_1$ and $y_2$ be the solutions of

$$\begin{equation}\label{eq:2.1E.12} y''+p(x)y'+q(x)y=0 \end{equation}$$

such that

$$\begin{eqnarray*} y_1(x_0)=1, \quad y'_1(x_0)=0\quad \mbox{and} \quad y_2(x_0)=0,\; y'_2(x_0)=1. \end{eqnarray*}$$

(Theorem $(2.1.1)$ implies that each of these initial value problems has a unique solution on $(a,b)$.)

(a) Show that $\{y_1,y_2\}$ is linearly independent on $(a,b)$.

(b) Show that an arbitrary solution $y$ of $\eqref{eq:2.1E.12}$ on $a,b)$ can be written as $y=y(x_0)y_1+y'(x_0)y_2$.

(c) Express the solution of the initial value problem

$$\begin{eqnarray*} y''+p(x)y'+q(x)y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1 \end{eqnarray*}$$

as a linear combination of $y_1$ and $y_2$.

Answer

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Exercise $\PageIndex{38}$

Find solutions $y_1$ and $y_2$ of the equation $y''=0$ that satisfy the initial conditions

$$\begin{eqnarray*} y_1(x_0)=1, \quad y'_1(x_0)=0 \quad \mbox{ and } \quad y_2(x_0)=0, \quad y'_2(x_0)=1. \end{eqnarray*}$$

Then use Exercise $(2.1E.37)$ (c) to write the solution of the initial value problem

$$\begin{eqnarray*} y''=0,\quad y(0)=k_0,\quad y'(0)=k_1 \end{eqnarray*}$$

as a linear combination of $y_1$ and $y_2$.

Answer

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Exercise $\PageIndex{39}$

Let $x_0$ be an arbitrary real number. Given Example $(2.1.1)$ that $e^x$ and $e^{-x}$ are solutions of $y''-y=0$, find solutions $y_1$ and $y_2$ of $y''-y=0$ such that

$$\begin{eqnarray*} y_1(x_0)=1, \quad y'_1(x_0)=0 \quad \mbox{and} \quad y_2(x_0)=0,\; y'_2(x_0)=1. \end{eqnarray*}$$

Then use Exercise $(2.1E.37)$ (c) to write the solution of the initial value problem

$$\begin{eqnarray*} y''-y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1 \end{eqnarray*}$$

as a linear combination of $y_1$ and $y_2$.

Answer

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Exercise $\PageIndex{40}$

Let $x_0$ be an arbitrary real number. Given Example $(2.1.2)$ that $\cos\omega x$ and $\sin\omega x$ are solutions of $y''+\omega^2y=0$, find solutions of $y''+\omega^2y=0$$ such that

$$\begin{eqnarray*} y_1(x_0)=1, \quad y'_1(x_0)=0 \quad \mbox{and} \quad y_2(x_0)=0,\; y'_2(x_0)=1. \end{eqnarray*}$$

Then use Exercise $(2.1E.37)$ (c) to write the solution of the initial value problem

$$\begin{eqnarray*} y''+\omega^2y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1 \end{eqnarray*}$$

as a linear combination of $y_1$ and $y_2$. Use the identities
$$\begin{eqnarray*} \cos(A+B)&=&\cos A\cos B-\sin A\sin B\\ \sin(A+B)&=&\sin A\cos B+\cos A\sin B \end{eqnarray*}$$

to simplify your expressions for $y_1$, $y_2$, and $y$.

Answer

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Exercise $\PageIndex{41}$

Recall from Exercise $(2.1E.4)$ that $1/(x-1)$ and $1/(x+1)$ are solutions of

$$\begin{equation}\label{eq:2.1E.13} (x^2-1)y''+4xy'+2y=0 \end{equation}$$

on $(-1,1)$. Find solutions of $\eqref{eq:2.1E.13}$ such that

$$\begin{eqnarray*} y_1(0)=1, \quad y'_1(0)=0 \quad \mbox{and} \quad y_2(0)=0,\; y'_2(0)=1. \end{eqnarray*}$$

Then use Exercise $(2.1E.37)$ (c) to write the solution of initial value problem

$$\begin{eqnarray*} (x^2-1)y''+4xy'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1 \end{eqnarray*}$$

as a linear combination of $y_1$ and $y_2$.

Answer

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Exercise $\PageIndex{42}$

(a) Verify that $y_1=x^2$ and $y_2=x^3$ satisfy

$$\begin{equation}\label{eq:2.1E.14} x^2y''-4xy'+6y=0 \end{equation}$$

on $(-\infty,\infty)$ and that $\{y_1,y_2\}$ is a fundamental set of solutions of $\eqref{eq:2.1E.14}$ on $(-\infty,0)$ and $(0,\infty)$.

(b) Let $a_1$, $a_2$, $b_1$, and $b_2$ be constants. Show that

$$\begin{eqnarray*} y=\left\{\begin{array}{rr} a_1x^2+a_2x^3,&x\ge 0,\\ b_1x^2+b_2x^3,&x<0\phantom{,} \end{array}\right. \end{eqnarray*}$$

is a solution of $\eqref{eq:2.1E.14}$ on $(-\infty,\infty)$ if and only if $a_1=b_1$. From this, justify the statement that $y$ is a solution of $\eqref{eq:2.1E.14}$ on $(-\infty,\infty)$ if and only if

$$\begin{eqnarray*} y=\left\{\begin{array}{rr} c_1x^2+c_2x^3,&x\ge 0,\\ c_1x^2+c_3x^3,&x<0, \end{array}\right. \end{eqnarray*}$$

where $c_1$, $c_2$, and $c_3$ are arbitrary constants.

(c) For what values of $k_0$ and $k_1$ does the initial value problem

$$\begin{eqnarray*} x^2y''-4xy'+6y=0,\quad y(0)=k_0,\quad y'(0)=k_1 \end{eqnarray*}$$

have a solution? What are the solutions?

(d) Show that if $x_0\ne0$ and $k_0,k_1$ are arbitrary constants, the initial value problem

$$\begin{equation}\label{eq:2.1E.15} x^2y''-4xy'+6y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1 \end{equation}$$

has infinitely many solutions on $(-\infty,\infty)$. On what interval does $\eqref{eq:2.1E.15}$ have a unique solution?

Answer

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Exercise $\PageIndex{43}$

(a) Verify that $y_1=x$ and $y_2=x^2$ satisfy

$$\begin{equation}\label{eq:2.1E.16} x^2y''-2xy'+2y=0 \end{equation}$$

on $(-\infty,\infty)$ and that $\{y_1,y_2\}$ is a fundamental set of solutions of $\eqref{eq:2.1E.16}$ on $(-\infty,0)$ and $(0,\infty)$.

(b) Let $a_1$, $a_2$, $b_1$, and $b_2$ be constants. Show that

$$\begin{eqnarray*} y=\left\{\begin{array}{rr} a_1x+a_2x^2,&x\ge 0,\\ b_1x+b_2x^2,&x<0\phantom{,} \end{array}\right. \end{eqnarray*}$$

is a solution of $\eqref{eq:2.1E.16}$ on $(-\infty,\infty)$ if and only if $a_1=b_1$ and $a_2=b_2$. From this, justify the statement that the general solution of $\eqref{eq:2.1E.16}$ on $(-\infty,\infty)$ is $y=c_1x+c_2x^2$, where $c_1$ and $c_2$ are arbitrary constants.

(c) For what values of $k_0$ and $k_1$ does the initial value problem

$$\begin{eqnarray*} x^2y''-2xy'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1 \end{eqnarray*}$$

have a solution? What are the solutions?

(d) Show that if $x_0\ne0$ and $k_0,k_1$ are arbitrary constants then the initial value problem

$$\begin{eqnarray*} x^2y''-2xy'+2y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1 \end{eqnarray*}$$

has a unique solution on $(-\infty,\infty)$.

Answer

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Exercise $\PageIndex{44}$

(a) Verify that $y_1=x^3$ and $y_2=x^4$ satisfy

$$\begin{equation}\label{eq:2.1E.17} x^2y''-6xy'+12y=0 \end{equation}$$

on $(-\infty,\infty)$, and that $\{y_1,y_2\}$ is a fundamental set of solutions of $\eqref{eq:2.1E.17}$ on $(-\infty,0)$ and $(0,\infty)$.

(b) Show that $y$ is a solution of $\eqref{eq:2.1E.17}$ on $(-\infty,\infty)$ if and only if

$$\begin{eqnarray*} y=\left\{\begin{array}{rr} a_1x^3+a_2x^4,&x\ge 0,\\ b_1x^3+b_2x^4,&x<0, \end{array}\right. \end{eqnarray*}$$

where $a_1$, $a_2$, $b_1$, and $b_2$ are arbitrary constants.

(c) For what values of $k_0$ and $k_1$ does the initial value problem

$$\begin{eqnarray*} x^2y''-6xy'+12y=0, \quad y(0)=k_0,\quad y'(0)=k_1 \end{eqnarray*}$$

have a solution? What are the solutions?

(d) Show that if $x_0\ne0$ and $k_0,k_1$ are arbitrary constants then the initial value problem

$$\begin{equation}\label{eq:2.1E.18} x^2y''-6xy'+12y=0, \quad y(x_0)=k_0,\quad y'(x_0)=k_1 \end{equation}$$

has infinitely many solutions on $(-\infty,\infty)$. On what interval does $\eqref{eq:2.1E.18}$ have a unique solution?

Answer

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