3.6E: Exercises

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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: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{2}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{3}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{4}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{5}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{6}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{7}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{8}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{9}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{10}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{11}\)

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

Answer: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{13}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{14}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{15}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{16}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{17}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{18}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{19}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{20}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{21}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{22}\)

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

Answer: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{24}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{25}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{26}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{27}\)

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

Answer: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{29}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{30}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{31}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{32}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{33}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{34}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{35}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{36}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{37}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{38}\)

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

Answer: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{40}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{41}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{42}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{43}\)

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

Answer: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{45}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{46}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{47}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{48}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{49}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{50}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{51}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{52}\)

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

Answer: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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*}

\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: Add texts here. Do not delete this text first.

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*}

\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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{59}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{60}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{61}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{62}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{63}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{64}\)

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

Answer: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{65}\)

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

Answer: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.