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# 3.6E: Exercises

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University

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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 $$\PageIndex{1}$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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 $$\PageIndex{12}$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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 $$\PageIndex{23}$$

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

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## Exercise $$\PageIndex{24}$$

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

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## Exercise $$\PageIndex{25}$$

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

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## Exercise $$\PageIndex{26}$$

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

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## Exercise $$\PageIndex{27}$$

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

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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 $$\PageIndex{28}$$

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

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## Exercise $$\PageIndex{29}$$

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

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## Exercise $$\PageIndex{30}$$

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

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## Exercise $$\PageIndex{31}$$

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

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## Exercise $$\PageIndex{32}$$

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

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## Exercise $$\PageIndex{33}$$

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

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## Exercise $$\PageIndex{34}$$

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

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## Exercise $$\PageIndex{35}$$

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

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## Exercise $$\PageIndex{36}$$

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

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## Exercise $$\PageIndex{37}$$

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

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## Exercise $$\PageIndex{38}$$

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

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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 $$\PageIndex{39}$$

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

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## Exercise $$\PageIndex{40}$$

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

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## Exercise $$\PageIndex{41}$$

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

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## Exercise $$\PageIndex{42}$$

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

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## Exercise $$\PageIndex{43}$$

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

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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 $$\PageIndex{44}$$

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

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## Exercise $$\PageIndex{45}$$

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

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## Exercise $$\PageIndex{46}$$

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

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## Exercise $$\PageIndex{47}$$

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

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## Exercise $$\PageIndex{48}$$

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

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## Exercise $$\PageIndex{49}$$

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

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## Exercise $$\PageIndex{50}$$

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

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## Exercise $$\PageIndex{51}$$

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

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## Exercise $$\PageIndex{52}$$

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

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## Exercise $$\PageIndex{53}$$

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

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

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

(a) Show that

\begin{eqnarray*}
\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 $$(3.6.2)$$.

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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*}
\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?)

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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*}
\end{eqnarray*}

Suppose

\begin{eqnarray*}
\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*}

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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.

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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*}
\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.

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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 $$\PageIndex{58}$$

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

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## Exercise $$\PageIndex{59}$$

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

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## Exercise $$\PageIndex{60}$$

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

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## Exercise $$\PageIndex{61}$$

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

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## Exercise $$\PageIndex{62}$$

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

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## Exercise $$\PageIndex{63}$$

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

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## Exercise $$\PageIndex{64}$$

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

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## Exercise $$\PageIndex{65}$$

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

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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*}
are solutions of $$Ly=0$$.