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Mathematics LibreTexts

3.6E: Exercises

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    17854
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    In Exercises \((3.6E.1)\) to \((3.6E.11)\), find a fundamental set of Frobenius solutions. Compute the terms involving \(x^{n+r_1}\), where \(0\le n\le N\) (\(N\) at least \(7\)) and \(r_1\) is the root of the indicial equation. Optionally, write a computer program to implement the applicable recurrence formulas and take \(N>7\).

    Exercise \(\PageIndex{1}\)

    \(x^2y''-x(1-x)y'+(1-x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{2}\)

    \(x^2(1+x+2x^2)y'+x(3+6x+7x^2)y'+(1+6x-3x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{3}\)

    \(x^2(1+2x+x^2)y''+x(1+3x+4x^2)y'-x(1-2x)y=0\)

    Answer

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    Exercise \(\PageIndex{4}\)

    \(4x^2(1+x+x^2)y''+12x^2(1+x)y'+(1+3x+3x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{5}\)

    \(x^2(1+x+x^2)y''-x(1-4x-2x^2)y'+y=0\)

    Answer

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    Exercise \(\PageIndex{6}\)

    \(9x^2y''+3x(5+3x-2x^2)y'+(1+12x-14x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{7}\)

    \(x^2y''+x(1+x+x^2)y'+x(2-x)y=0\)

    Answer

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    Exercise \(\PageIndex{8}\)

    \(x^2(1+2x)y''+x(5+14x+3x^2)y'+(4+18x+12x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{9}\)

    \(4x^2y''+2x(4+x+x^2)y'+(1+5x+3x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{10}\)

    \(16x^2y''+4x(6+x+2x^2)y'+(1+5x+18x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{11}\)

    \(9x^2(1+x)y''+3x(5+11x-x^2)y'+(1+16x-7x^2)y=0\)

    Answer

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    In Exercises \((3.6E.12)\) to \((3.6E.22)\), find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients.

    Exercise \(\PageIndex{12}\)

    \(4x^2y''+(1+4x)y=0\)

    Answer

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    Exercise \(\PageIndex{13}\)

    \(36x^2(1-2x)y''+24x(1-9x)y'+(1-70x)y=0\)

    Answer

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    Exercise \(\PageIndex{14}\)

    \(x^2(1+x)y''-x(3-x)y'+4y=0\)

    Answer

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    Exercise \(\PageIndex{15}\)

    \(x^2(1-2x)y''-x(5-4x)y'+(9-4x)y=0\)

    Answer

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    Exercise \(\PageIndex{16}\)

    \(25x^2y''+x(15+x)y'+(1+x)y=0\)

    Answer

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    Exercise \(\PageIndex{17}\)

    \(2x^2(2+x)y''+x^2y'+(1-x)y=0\)

    Answer

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    Exercise \(\PageIndex{18}\)

    \(x^2(9+4x)y''+3xy'+(1+x)y=0\)

    Answer

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    Exercise \(\PageIndex{19}\)

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

    Answer

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    Exercise \(\PageIndex{20}\)

    \(x^2(1-4x)y''+3x(1-6x)y'+(1-12x)y=0\)

    Answer

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    Exercise \(\PageIndex{21}\)

    \(x^2(1+2x)y''+x(3+5x)y'+(1-2x)y=0\)

    Answer

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    Exercise \(\PageIndex{22}\)

    \(2x^2(1+x)y''-x(6-x)y'+(8-x)y=0\)

    Answer

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    In Exercises \((3.6E.23)\) to \((3.6E.27)\), find a fundamental set of Frobenius solutions. Compute the terms involving \(x^{n+r_1}\), where \(0\le n\le N\) (\(N\) at least \(7\)) and \(r_1\) is the root of the indicial equation. Optionally, write a computer program to implement the applicable recurrence formulas and take \(N>7\).

    Exercise \(\PageIndex{23}\)

    \(x^2(1+2x)y''+x(5+9x)y'+(4+3x)y=0\)

    Answer

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    Exercise \(\PageIndex{24}\)

    \(x^2(1-2x)y''-x(5+4x)y'+(9+4x)y=0\)

    Answer

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    Exercise \(\PageIndex{25}\)

    \(x^2(1+4x)y''-x(1-4x)y'+(1+x)y=0\)

    Answer

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    Exercise \(\PageIndex{26}\)

    \(x^2(1+x)y''+x(1+2x)y'+xy=0\)

    Answer

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    Exercise \(\PageIndex{27}\)

    \(x^2(1-x)y''+x(7+x)y'+(9-x)y=0\)

    Answer

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    In Exercises \((3.6E.28)\) to \((3.6E.38)\), find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients.

    Exercise \(\PageIndex{28}\)

    \(x^2y''-x(1-x^2)y'+(1+x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{29}\)

    \(x^2(1+x^2)y''-3x(1-x^2)y'+4y=0\)

    Answer

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    Exercise \(\PageIndex{30}\)

    \(4x^2y''+2x^3y'+(1+3x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{31}\)

    \(x^2(1+x^2)y''-x(1-2x^2)y'+y=0\)

    Answer

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    Exercise \(\PageIndex{32}\)

    \(2x^2(2+x^2)y''+7x^3y'+(1+3x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{33}\)

    \(x^2(1+x^2)y''-x(1-4x^2)y'+(1+2x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{34}\)

    \(4x^2(4+x^2)y''+3x(8+3x^2)y'+(1-9x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{35}\)

    \(3x^2(3+x^2)y''+x(3+11x^2)y'+(1+5x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{36}\)

    \(4x^2(1+4x^2)y''+32x^3y'+y=0\)

    Answer

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    Exercise \(\PageIndex{37}\)

    \(9x^2y''-3x(7-2x^2)y'+(25+2x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{38}\)

    \(x^2(1+2x^2)y''+x(3+7x^2)y'+(1-3x^2)y=0\)

    Answer

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    In Exercises \((3.6E.39)\) to \((3.6E.43)\), find a fundamental set of Frobenius solutions. Compute the terms involving \(x^{2m+r_1}\), where \(0\le m\le M\) (\(M\) at least \(3\)) and \(r_1\) is the root of the indicial equation. Optionally, write a computer program to implement the applicable recurrence formulas and take \(M>3\).

    Exercise \(\PageIndex{39}\)

    \(x^2(1+x^2)y''+x(3+8x^2)y'+(1+12x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{40}\)

    \(x^2y''-x(1-x^2)y'+(1+x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{41}\)

    \(x^2(1-2x^2)y''+x(5-9x^2)y'+(4-3x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{42}\)

    \(x^2(2+x^2)y''+x(14-x^2)y'+2(9+x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{43}\)

    \(x^2(1+x^2)y''+x(3+7x^2)y'+(1+8x^2)y=0\)

    Answer

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    In Exercises \((3.6E.44)\) to \((3.6E.52)\), find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients.

    Exercise \(\PageIndex{44}\)

    \(x^2(1-2x)y''+3xy'+(1+4x)y=0\)

    Answer

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    Exercise \(\PageIndex{45}\)

    \(x(1+x)y''+(1-x)y'+y=0\)

    Answer

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    Exercise \(\PageIndex{46}\)

    \(x^2(1-x)y''+x(3-2x)y'+(1+2x)y=0\)

    Answer

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    Exercise \(\PageIndex{47}\)

    \(4x^2(1+x)y''-4x^2y'+(1-5x)y=0\)

    Answer

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    Exercise \(\PageIndex{48}\)

    \(x^2(1-x)y''-x(3-5x)y'+(4-5x)y=0\)

    Answer

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    Exercise \(\PageIndex{49}\)

    \(x^2(1+x^2)y''-x(1+9x^2)y'+(1+25x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{50}\)

    \(9x^2y''+3x(1-x^2)y'+(1+7x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{51}\)

    \(x(1+x^2)y''+(1-x^2)y'-8xy=0\)

    Answer

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    Exercise \(\PageIndex{52}\)

    \(4x^2y''+2x(4-x^2)y'+(1+7x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{53}\)

    Under the assumptions of Theorem \((3.6.2)\), suppose the power series

    \begin{eqnarray*}
    \sum_{n=0}^\infty a_n(r_1)x^n \quad\mbox{ and }\quad \sum_{n=1}^\infty a_n'(r_1)x^n
    \end{eqnarray*}

    converge on \((-\rho,\rho)\).

    (a) Show that

    \begin{eqnarray*}
    y_1=x^{r_1}\sum_{n=0}^\infty a_n(r_1)x^n\quad\mbox{ and }\quad y_2=y_1\ln x+x^{r_1}\sum_{n=1}^\infty a_n'(r_1)x^n
    \end{eqnarray*}

    are linearly independent on \((0,\rho)\).

    Hint: Show that if \(c_1\) and \(c_2\) are constants such that \(c_1y_1+c_2y_2\equiv0\) on \((0,\rho)\), then

    \begin{eqnarray*}
    (c_1+c_2\ln x)\sum_{n=0}^\infty a_n(r_1)x^n+ c_2\sum_{n=1}^\infty a_n'(r_1)x^n=0,\quad 0<x<\rho.
    \end{eqnarray*}

    Then let \(x\to0+\) to conclude that \(c_2=0\).

    (b) Use the result of part (a) to complete the proof of Theorem \((3.6.2)\).

    Answer

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    Exercise \(\PageIndex{54}\)

    Let

    \begin{eqnarray*}
    Ly=x^2(\alpha_0+\alpha_1x)y''+x(\beta_0+\beta_1x)y'+(\gamma_0+\gamma_1x)y
    \end{eqnarray*}

    and define

    \begin{eqnarray*}
    p_0(r)=\alpha_0r(r-1)+\beta_0r+\gamma_0\quad\mbox{ and }\quad p_1(r)=\alpha_1r(r-1)+\beta_1r+\gamma_1.
    \end{eqnarray*}

    Theorem \((3.6.1)\) and Exercise \((3.5E.55)\) part (a) imply that if

    \begin{eqnarray*}
    y(x,r)=x^r\sum_{n=0}^\infty a_n(r)x^n
    \end{eqnarray*}

    where

    \begin{eqnarray*}
    a_n(r)=(-1)^n\prod_{j=1}^n{p_1(j+r-1)\over p_0(j+r)},
    \end{eqnarray*}

    then

    \begin{eqnarray*}
    Ly(x,r)=p_0(r)x^r.
    \end{eqnarray*}

    Now suppose \(p_0(r)=\alpha_0(r-r_1)^2\) and \(p_1(k+r_1)\ne0\) if \(k\) is a nonnegative integer.

    (a) Show that \(Ly=0\) has the solution

    \begin{eqnarray*}
    y_1=x^{r_1}\sum_{n=0}^\infty a_n(r_1)x^n,
    \end{eqnarray*}

    where

    \begin{eqnarray*}
    a_n(r_1)={(-1)^n\over\alpha_0^n(n!)^2}\prod_{j=1}^np_1(j+r_1-1).
    \end{eqnarray*}

    (b) Show that \(Ly=0\) has the second solution

    \begin{eqnarray*}
    y_2=y_1\ln x+x^{r_1}\sum_{n=1}^\infty a_n(r_1)J_nx^n,
    \end{eqnarray*}

    where

    \begin{eqnarray*}
    J_n=\sum_{j=1}^n{p_1'(j+r_1-1)\over p_1(j+r_1-1)}-2\sum_{j=1}^n{1\over j}.
    \end{eqnarray*}

    (c) Conclude from part (a) and part (b) that if \(\gamma_1\ne0\) then

    \begin{eqnarray*}
    y_1=x^{r_1}\sum_{n=0}^\infty {(-1)^n\over(n!)^2}\left(\gamma_1\over\alpha_0\right)^nx^n
    \end{eqnarray*}

    and

    \begin{eqnarray*}
    y_2=y_1\ln x-2x^{r_1}\sum_{n=1}^\infty {(-1)^n\over(n!)^2}\left(\gamma_1\over\alpha_0\right)^n \left(\sum_{j=1}^n{1\over j}\right)x^n
    \end{eqnarray*}

    are solutions of

    \begin{eqnarray*}
    \alpha_0x^2y''+\beta_0xy'+(\gamma_0+\gamma_1x)y=0.
    \end{eqnarray*}

    (The conclusion is also valid if \(\gamma_1=0\). Why?)

    Answer

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    Exercise \(\PageIndex{55}\)

    Let

    \begin{eqnarray*}
    Ly=x^2(\alpha_0+\alpha_qx^q)y''+x(\beta_0+\beta_qx^q)y'+(\gamma_0+\gamma_qx^q)y
    \end{eqnarray*}

    where \(q\) is a positive integer, and define

    \begin{eqnarray*}
    p_0(r)=\alpha_0r(r-1)+\beta_0r+\gamma_0\quad\mbox{ and }\quad p_q(r)=\alpha_qr(r-1)+\beta_qr+\gamma_q.
    \end{eqnarray*}

    Suppose

    \begin{eqnarray*}
    p_0(r)=\alpha_0(r-r_1)^2 \quad\mbox{ and }\quad p_q(r)\not\equiv0.
    \end{eqnarray*}

    (a) Recall from Exercise \((3.5E.59)\) that \(Ly=0\) has the solution

    \begin{eqnarray*}
    y_1=x^{r_1}\sum_{m=0}^\infty a_{qm}(r_1)x^{qm},
    \end{eqnarray*}

    where

    \begin{eqnarray*}
    a_{qm}(r_1)={(-1)^m\over (q^2\alpha_0)^m(m!)^2}\prod_{j=1}^mp_q\left(q(j-1)+r_1\right).
    \end{eqnarray*}

    (b) Show that \(Ly=0\) has the second solution

    \begin{eqnarray*}
    y_2=y_1\ln x+x^{r_1}\sum_{m=1}^\infty a_{qm}'(r_1)J_mx^{qm},
    \end{eqnarray*}

    where

    \begin{eqnarray*}
    J_m=\sum_{j=1}^m{p_q'\left(q(j-1)+r_1\right)\over p_q\left(q(j-1)+r_1\right)}-{2\over q}\sum_{j=1}^m{1\over j}.
    \end{eqnarray*}

    (c) Conclude from part (a) and part (b) that if \(\gamma_q\ne0\) then

    \begin{eqnarray*}
    y_1=x^{r_1}\sum_{m=0}^\infty {(-1)^m\over(m!)^2}\left(\gamma_q\over q^2\alpha_0\right)^mx^{qm}
    \end{eqnarray*}

    and

    \begin{eqnarray*}
    y_2=y_1\ln x-{2\over q}x^{r_1}\sum_{m=1}^\infty {(-1)^m\over(m!)^2}\left(\gamma_q\over q^2\alpha_0\right)^m\left(\sum_{j=1}^m{1\over j}\right)x^{qm}
    \end{eqnarray*}

    are solutions of

    \begin{eqnarray*}
    \alpha_0x^2y''+\beta_0xy'+(\gamma_0+\gamma_qx^q)y=0.
    \end{eqnarray*}

    Answer

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    Exercise \(\PageIndex{56}\)

    The equation

    \begin{eqnarray*}
    xy''+y'+xy=0
    \end{eqnarray*}

    is Bessel's equation of order zero. (See Exercise \((3.5E.53)\).) Find two linearly independent Frobenius solutions of this equation.

    Answer

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    Exercise \(\PageIndex{57}\)

    Suppose the assumptions of Exercise \((3.5E.53)\) hold, except that

    \begin{eqnarray*}
    p_0(r)=\alpha_0(r-r_1)^2.
    \end{eqnarray*}

    Show that

    \begin{eqnarray*}
    y_1={x^{r_1}\over\alpha_0+\alpha_1x+\alpha_2x^2} \quad \mbox{and}\quad y_2={x^{r_1}\ln x\over\alpha_0+\alpha_1x+\alpha_2x^2}
    \end{eqnarray*}

    are linearly independent Frobenius solutions of

    \begin{eqnarray*}
    x^2(\alpha_0+\alpha_1x+\alpha_2 x^2)y''+x(\beta_0+\beta_1x+\beta_2x^2)y'+ (\gamma_0+\gamma_1x+\gamma_2x^2)y=0
    \end{eqnarray*}

    on any interval \((0,\rho)\) on which \(\alpha_0+\alpha_1x+\alpha_2x^2\) has no zeros.

    Answer

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    In Exercises \((3.6E.58)\) to \((3.6E.65)\), use the method suggested by Exercise \((3.6E.57)\) to find the general solution on some interval \((0,\rho)\).

    Exercise \(\PageIndex{58}\)

    \(4x^2(1+x)y''+8x^2y'+(1+x)y=0\)

    Answer

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    Exercise \(\PageIndex{59}\)

    \(9x^2(3+x)y''+3x(3+7x)y'+(3+4x)y=0\)

    Answer

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    Exercise \(\PageIndex{60}\)

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

    Answer

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    Exercise \(\PageIndex{61}\)

    \(16x^2(1+x^2)y''+8x(1+9x^2)y'+(1+49x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{62}\)

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

    Answer

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    Exercise \(\PageIndex{63}\)

    \(4x^2(1+3x+x^2)y''+8x^2(3+2x)y'+(1+3x+9x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{64}\)

    \(x^2(1-x)^2y''-x(1+2x-3x^2)y'+(1+x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{65}\)

    \(9x^2(1+x+x^2)y''+3x(1+7x+13x^2)y'+(1+4x+25x^2)y=0\)

    Answer

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    Exercise \(\PageIndex{66}\)

    (a) Let \(L\) and \(y(x,r)\) be as in Exercises \((3.5E.57)\) and \((3.5E.58)\). Extend Theorem \((3.6.1)\) by showing that

    \begin{eqnarray*}
    L\left({\partial y\over \partial r}(x,r)\right)=p'_0(r)x^r+x^rp_0(r)\ln x.
    \end{eqnarray*}

    (b) Show that if

    \begin{eqnarray*}
    p_0(r)=\alpha_0(r-r_1)^2
    \end{eqnarray*}

    then

    \begin{eqnarray*}
    y_1=y(x,r_1) \quad \mbox{and} \quad y_2={\partial y\over\partial r}(x,r_1)
    \end{eqnarray*}

    are solutions of \(Ly=0\).

    Answer

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