
# 2.1E: Exercises

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

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

\label{eq:2.1.1}
y''-7y'+10y=0

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*}
\end{eqnarray*}

(d) Solve the initial value problem

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

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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

\label{eq:2.1.2}
y''-2y'+2y=0

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*}
\end{eqnarray*}

(d) Solve the initial value problem

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

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

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

\label{eq:2.1.3}
y''-2y'+y=0

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*}
\end{eqnarray*}

(d) Solve the initial value problem

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

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

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

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

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*}
\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 $$x_0=0$$.

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

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

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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.)

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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.)

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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

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

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

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

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 $$(a,b)$$.

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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 $$\PageIndex{10}$$

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

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

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

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

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

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

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

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

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

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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\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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*}
\end{eqnarray*}

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

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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*}
\end{eqnarray*}

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

(b) The initial value problem

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

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

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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

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

on $$(a,b)$$. Let

\begin{eqnarray*}
\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 $$\{y_1,y_2\}$$.

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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

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

on $$(a,b)$$. Let

\begin{eqnarray*}
\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 $$\alpha\gamma-\beta\delta\ne0$$.

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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 $$(a,b)$$.

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

Let

\begin{eqnarray*}
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 $$(a,b)$$.

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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 $$(a,b)$$.

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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.

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

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

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

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 $$(a,b)$$.

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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*}
\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.

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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

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

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.

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

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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

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

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 $$(a,b)$$.

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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

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

such that

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

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

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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*}
\end{eqnarray*}

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

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

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

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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*}
\end{eqnarray*}

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

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

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

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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*}
\end{eqnarray*}

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

\begin{eqnarray*}
\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 $$y$$.

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

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

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

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

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

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

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

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

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

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

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

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*}
\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

\label{eq:2.1E.15}

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

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

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

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

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*}
\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*}
\end{eqnarray*}

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

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## Exercise $$\PageIndex{44}$$

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

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

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*}
\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

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