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# 3.4E: Exercises

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University
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In Exercises $$(3.4E.1)$$ to $$(3.4E.18)$$, find the general solution of the given Euler equation on $$(0,\infty)$$.

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

$$x^2y''+7xy'+8y=0$$

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

$$x^2y''-7xy'+7y=0$$

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

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

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

$$x^2y''+5xy'+4y=0$$

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

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

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

$$x^2y''-3xy'+13y=0$$

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

$$x^2y''+3xy'-3y=0$$

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

$$12x^2y''-5xy''+6y=0$$

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

$$4x^2y''+8xy'+y=0$$

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

$$3x^2y''-xy'+y=0$$

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

$$2x^2y''-3xy'+2y=0$$

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

$$x^2y''+3xy'+5y=0$$

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

$$9x^2y''+15xy'+y=0$$

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

$$x^2y''-xy'+10y=0$$

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

$$x^2y''-6y=0$$

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

$$2x^2y''+3xy'-y=0$$

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

$$x^2y''-3xy'+4y=0$$

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

$$2x^2y''+10xy'+9y=0$$

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

(a) Adapt the proof of Theorem $$(3.4.3)$$ to show that $$y=y(x)$$ satisfies the Euler equation

\begin{equation}\label{eq:3.4E.1}
ax^2y''+bxy'+cy=0
\end{equation}

on $$(-\infty,0)$$ if and only if $$Y(t)=y(-e^t)$$

\begin{eqnarray*}
a {d^2Y\over dt^2}+(b-a){dY\over dt}+cY=0.
\end{eqnarray*}

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

(b) Use part (a) to show that the general solution of \eqref{eq:3.4E.1} on $$(-\infty,0)$$ is

\begin{eqnarray*}
y&=&c_1|x|^{r_1}+c_2|x|^{r_2}\mbox{ if $$r_1$$ and $$r_2$$ are distinct real numbers; }\\
y&=&|x|^{r_1}(c_1+c_2\ln|x|)\mbox{ if $$r_1=r_2$$; }\\
y&=&|x|^{\lambda}\left[c_1\cos\left(\omega\ln|x|\right)+ c_2\sin\left(\omega\ln|x| \right)\right]\mbox{ if $$r_1,r_2=\lambda\pm i\omega$$ with $$\omega>0$$}.
\end{eqnarray*}

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

Use reduction of order to show that if

\begin{eqnarray*}
ar(r-1)+br+c=0
\end{eqnarray*}

has a repeated root $$r_1$$ then $$y=x^{r_1}(c_1+c_2\ln x)$$ is the general solution of

\begin{eqnarray*}
ax^2y''+bxy'+cy=0
\end{eqnarray*}

on $$(0,\infty)$$.

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

A nontrivial solution of

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

is said to be $$\textcolor{blue}{\mbox{oscillatory}}$$ on an interval $$(a,b)$$ if it has infinitely many zeros on $$(a,b)$$. Otherwise $$y$$ is said to be $$\textcolor{blue}{\mbox{nonoscillatory}}$$ on $$(a,b)$$. Show that the equation

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

has oscillatory solutions on $$(0,\infty)$$ if and only if $$k>1/4$$.

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

In Example $$(3.4.2)$$ we saw that $$x_0=1$$ and $$x_0=-1$$ are regular singular points of Legendre's equation

\begin{equation}\label{eq:3.4E.2}
(1-x^2)y''-2xy'+\alpha(\alpha+1)y=0.
\end{equation}

(a) Introduce the new variables $$t=x-1$$ and $$Y(t)=y(t+1)$$, and show that $$y$$ is a solution of \eqref{eq:3.4E.2} if and only if $$Y$$ is a solution of

\begin{eqnarray*}
t(2+t){d^2Y\over dt^2}+2(1+t){dY\over dt}-\alpha(\alpha+1)Y=0,
\end{eqnarray*}

which has a regular singular point at $$t_0=0$$.

(b) Introduce the new variables $$t=x+1$$ and $$Y(t)=y(t-1)$$, and show that $$y$$ is a solution of \eqref{eq:3.4E.2} if and only if $$Y\( is a solution of \begin{eqnarray*} t(2-t){d^2Y\over dt^2}+2(1-t){dY\over dt}+\alpha(\alpha+1)Y=0, \end{eqnarray*} which has a regular singular point at \(t_0=0$$.

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

Let $$P_0,P_1$$, and $$P_2$$ be polynomials with no common factor, and suppose $$x_0\ne0$$ is a singular point of

\begin{equation}\label{eq:3.4E.3}
P_0(x)y''+P_1(x)y'+P_2(x)y=0.
\end{equation}

Let $$t=x-x_0$$ and $$Y(t)=y(t+x_0)$$.

(a) Show that $$y$$ is a solution of \eqref{eq:3.4E.3} if and only if $$Y$$ is a solution of

\begin{equation}\label{eq:3.4E.4}
R_0(t){d^2Y\over dt^2}+R_1(t){dY\over dt}+R_2(t)Y=0.
\end{equation}

where

\begin{eqnarray*}
(b) Show that $$R_0$$, $$R_1$$, and $$R_2$$ are polynomials in $$t$$ with no common factors, and $$R_0(0)=0$$; thus, $$t_0=0$$ is a singular point of \eqref{eq:3.4E.4}.