Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

3.4E: Exercises

  • Page ID
    17626
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    In Exercises \((3.4E.1)\) to \((3.4E.18)\), find the general solution of the given Euler equation on \((0,\infty)\).

    Exercise \(\PageIndex{1}\)

    \(x^2y''+7xy'+8y=0\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{2}\)

    \(x^2y''-7xy'+7y=0\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{3}\)

    \(x^2y''-xy'+y=0\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{4}\)


    \(x^2y''+5xy'+4y=0\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{5}\)

    \(x^2y''+xy'+y=0\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{6}\)

    \(x^2y''-3xy'+13y=0\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{7}\)

    \(x^2y''+3xy'-3y=0\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{8}\)

    \(12x^2y''-5xy''+6y=0\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{9}\)

    \(4x^2y''+8xy'+y=0\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{10}\)

    \(3x^2y''-xy'+y=0\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{11}\)

    \(2x^2y''-3xy'+2y=0\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{12}\)

    \(x^2y''+3xy'+5y=0\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{13}\)

    \(9x^2y''+15xy'+y=0\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{14}\)

    \(x^2y''-xy'+10y=0\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{15}\)

    \(x^2y''-6y=0\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{16}\)

    \(2x^2y''+3xy'-y=0\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{17}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{18}\)

    \(2x^2y''+10xy'+9y=0\)

    Answer

    Add texts here. Do not delete this text first.

    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*}

    Answer

    Add texts here. Do not delete this text first.

    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

    Add texts here. Do not delete this text first.

    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

    Add texts here. Do not delete this text first.

    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

    Add texts here. Do not delete this text first.

    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

    Add texts here. Do not delete this text first.