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# 3.7E: Exercises

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University
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In Exercises $$(3.7E.1)$$ to $$(3.7E.40)$$, find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients.

## Exercise $$\PageIndex{1}$$

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

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

$$xy''+y=0$$

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

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

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

$$xy''+xy'+y=0$$

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

$$2x^2(2+3x)y''+x(4+21x)y'-(1-9x)y=0$$

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

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

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

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

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

$$x^2y''+10xy'+(14+x)y=0$$

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

$$4x^2(1+x)y''+4x(3+8x)y'-(5-49x)y=0$$

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

$$x^2(1+x)y''-x(3+10x)y'+30xy=0$$

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

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

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## Exercise $$\PageIndex{12}$$

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

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

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

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

$$x^2(1+2x)y''+x(9+13x)y'+(7+5x)y=0$$

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

$$4x^2y''-2x(4-x)y'-(7+5x)y=0$$

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

$$3x^2(3+x)y''-x(15+x)y'-20y=0$$

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

$$x^2(1+x)y''+x(1-10x)y'-(9-10x)y=0$$

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

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

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

$$x^2(1+2x)y''-2x(3+14x)y'+(6+100x)y=0$$

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

$$x^2(1+x)y''-x(6+11x)y'+(6+32x)y=0$$

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

$$4x^2(1+x)y''+4x(1+4x)y'-(49+27x)y=0$$

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

$$x^2(1+2x)y''-x(9+8x)y'-12xy=0$$

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## Exercise $$\PageIndex{23}$$

$$x^2(1+x^2)y''-x(7-2x^2)y'+12y=0$$

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

$$x^2y''-x(7-x^2)y'+12y=0$$

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

$$xy''-5y'+xy=0$$

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

$$x^2y''+x(1+2x^2)y'-(1-10x^2)y=0$$

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

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

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## Exercise $$\PageIndex{28}$$

$$4x^2y''+2x(8+x^2)y'+(5+3x^2)y=0$$

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

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

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

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

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

$$4x^2y''+8xy'-(35-x^2)y=0$$

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

$$9x^2y''-3x(11+2x^2)y'+(13+10x^2)y=0$$

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

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

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

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

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

$$x^2(1+x^2)y''+x(5+11x^2)y'+24x^2y=0$$

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

$$4x^2(1+x^2)y''+8xy'-(35-x^2)y=0$$

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

$$x^2(1+x^2)y''-x(5-x^2)y'-(7+25x^2)y=0$$

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

$$x^2(1+x^2)y''+x(5+2x^2)y'-21y=0$$

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## Exercise $$\PageIndex{39}$$

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

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

$$4x^2(1+x^2)y''+4x(2+x^2)y'-(15+x^2)y=0$$

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

(a) Under the assumptions of Theorem $$(3.7.1)$$, show that

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

and

\begin{eqnarray*}
y_2=x^{r_2}\sum_{n=0}^{k-1}a_n(r_2)x^n+C\left(y_1\ln x+x^{r_1}\sum_{n=1}^\infty a_n'(r_1)x^n\right)
\end{eqnarray*}

are linearly independent.

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

\begin{eqnarray*}
\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 $$(3.7.1)$$.

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

Find a fundamental set of Frobenius solutions of Bessel's equation

\begin{eqnarray*}
x^2y''+xy'+(x^2-\nu^2)y=0
\end{eqnarray*}

in the case where $$\nu$$ is a positive integer.

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

Prove Theorem $$(3.7.2)$$.

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## Exercise $$\PageIndex{44}$$

Under the assumptions of Theorem $$(3.7.1)$$, show that $$C=0$$ if and only if $$p_1(r_2+ \ell)=0$$ for some integer $$\ell$$ in $$\{0,1,\dots,k-1\}$$.

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

Under the assumptions of Theorem $$(3.7.2)$$, show that $$C=0$$ if and only if $$p_2(r_2+2\ell)=0$$ for some integer $$\ell$$ in $$\{0,1,\dots,k-1\}$$.

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

Let

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

and define

\begin{eqnarray*}
p_0(r)=\alpha_0r(r-1)+\beta_0r+\gamma_0.
\end{eqnarray*}

Show that if

\begin{eqnarray*}
p_0(r)=\alpha_0(r-r_1)(r-r_2)
\end{eqnarray*}

where $$r_1-r_2=k$$, a positive integer, then $$Ly=0$$ has the solutions

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

and

\begin{eqnarray*}
y_2&=&x^{r_2}\sum_{n=0}^{k-1} {(-1)^n\over n!\prod_{j=1}^n(j-k)} \left(\gamma_1\over\alpha_0\right)^n x^n\\
&&-{1\over k!(k-1)!}\left(\gamma_1\over\alpha_0\right)^k\left(y_1\ln x- x^{r_1}\sum_{n=1}^\infty {(-1)^n\over n!\prod_{j=1}^n(j+k)}\left(\gamma_1\over\alpha_0\right)^n \left(\sum_{j=1}^n{2j+k\over j(j+k)}\right)x^n\right).
\end{eqnarray*}

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

Let

\begin{eqnarray*}
Ly=\alpha_0x^2y''+\beta_0xy'+(\gamma_0+\gamma_2x^2)y
\end{eqnarray*}

and define

\begin{eqnarray*}
p_0(r)=\alpha_0r(r-1)+\beta_0r+\gamma_0.
\end{eqnarray*}

Show that if

\begin{eqnarray*}
p_0(r)=\alpha_0(r-r_1)(r-r_2)
\end{eqnarray*}

where $$r_1-r_2=2k$$, an even positive integer, then $$Ly=0$$ has the solutions

\begin{eqnarray*}
y_1=x^{r_1}\sum_{m=0}^\infty {(-1)^m\over 4^mm!\prod_{j=1}^m(j+k)}\left(\gamma_2\over\alpha_0\right)^m x^{2m}
\end{eqnarray*}

and

\begin{eqnarray*}
y_2&=&x^{r_2}\sum_{m=0}^{k-1} {(-1)^m\over4^mm!\prod_{j=1}^m(j-k)} \left(\gamma_2\over\alpha_0\right)^m x^{2m}\\
&&-{2\over 4^kk!(k-1)!}\left(\gamma_2\over\alpha_0\right)^k\left(y_1\ln x- {x^{r_1}\over2}\sum_{m=1}^\infty {(-1)^m\over 4^mm!\prod_{j=1}^m(j+k)}\left(\gamma_2\over\alpha_0\right)^m \left(\sum_{j=1}^m{2j+k\over j(j+k)}\right)x^{2m}\right).
\end{eqnarray*}

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

Let $$L$$ be as in Exercises $$(3.5E.57)$$ and $$(3.5E.58)$$, and suppose the indicial polynomial of $$Ly=0$$ is

\begin{eqnarray*}
p_0(r)=\alpha_0(r-r_1)(r-r_2),
\end{eqnarray*}

with $$k=r_1-r_2$$, where $$k$$ is a positive integer. Define $$a_0(r)=1$$ for all $$r$$. If $$r$$ is a real number such that $$p_0(n+r)$$
is nonzero for all positive integers $$n$$, define

\begin{eqnarray*}
a_n(r)=-{1\over p_0(n+r)}\sum_{j=1}^n p_j(n+r-j)a_{n-j}(r),\,n\ge1,
\end{eqnarray*}

and let

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

Define

\begin{eqnarray*}
a_n(r_2)=-{1\over p_0(n+r_2)}\sum_{j=1}^n p_j(n+r_2-j)a_{n-j}(r_2)\mbox{ if } n\ge1\mbox{ and }n\ne k,
\end{eqnarray*}

and let $$a_k(r_2)$$ be arbitrary.

(a) Conclude from Exercise $$(3.6E.66)$$ that

\begin{eqnarray*}
L\left(y_1\ln x+x^{r_1}\sum_{n=1}^\infty a_n'(r_1)x^n\right)=k\alpha_0x^{r_1}.
\end{eqnarray*}

(b) Conclude from Exercise $$(3.5E.57)$$ that

\begin{eqnarray*}
L\left(x^{r_2}\sum_{n=0}^\infty a_n(r_2)x^n\right)=Ax^{r_1},
\end{eqnarray*}

where

\begin{eqnarray*}
A=\sum_{j=1}^k p_j(r_1-j)a_{k-j}(r_2).
\end{eqnarray*}

(c) Show that $$y_1$$ and

\begin{eqnarray*}
y_2=x^{r_2}\sum_{n=0}^\infty a_n(r_2)x^n -{A\over k\alpha_0} \left(y_1 \ln x+x^{r_1}\sum_{n=1}^\infty a_n'(r_1)x^n\right)
\end{eqnarray*}

form a fundamental set of Frobenius solutions of $$Ly=0$$.

(d) Show that choosing the arbitrary quantity $$a_k(r_2)$$ to be nonzero merely adds a multiple of $$y_1$$ to $$y_2$$. Conclude that we may as well take $$a_k(r_2)=0$$.