# 3.4E: Exercises

- Page ID
- 17626

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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 \((0,\infty)\).

**Answer**-
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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*}

x^2y''+ky=0 \quad (k=\; \mbox{constant})

\end{eqnarray*}

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

**Answer**-
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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 \(t_0=0\).

**Answer**-
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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*}

R_i(t)=P_i(t+x_0),\quad i=0,1,2.

\end{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}.

**Answer**-
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