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QUESTION

Obtain at least one solution of the form $$
y=x^{\sigma}\sum_{n=0}^{\infty}a_nx^n $$ for each of the following
differential equations. Where possible, obtain a second
independent solution of the same form, or comment on why it is not
possible to do so.

$x^2y''+xy'+\left(x^2-p^2\right)y=0$

Discuss the cases (a) $2p$ not an integer, (b) $2p$ an integer but
$p$ not an integer and (c) $p$ an integer, separately.

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ANSWER

$x^2y''+xy'+\left(x^2-p^2\right)y=0$ This is the Bessel equation.
0 is a
 regular singular point.\\
 $\ds \sum_{n=0}^\infty
 a_n\{\left[\left(\sigma+n\right)\left(\sigma+n-1\right)
 +\left(\sigma+n\right)-p^2\right]x^{\sigma+n}+x^{\sigma+n+2}\}=0$\\
 Factorizing\\
 $\ds \sum_{n=0}^\infty\{a_n\left[\left(\sigma+n\right)^2-p^2\right]
 x^{\sigma+n}+x^{\sigma +n+2}\}=0$\\
 Reordering\\
 $a_0\left(\sigma^2-p^2\right)x^\sigma+a_1
 \left[\left(\sigma+1\right)^2-p^2\right]x^{\sigma+1}$

 $+\ds \sum_{n=2}^\infty
 x^{\sigma +n}\{a_n\left[\left(\sigma+n\right)^2-p^2\right]+a_{n-2}\}=0\\
 a_0 \neq 0$ gives $\sigma^2-p^2=0 \Rightarrow \sigma=\pm
 p\Rightarrow a_1=0$ (If we assume $a_0=0$ and $a_1 \neq 0$, we get
 the same solution, only written differently. As a convention, we
 fix $\sigma$ by assuming $a_0 \neq 0$)\\
 $a_n=-\frac{1}{\left(n \pm p\right)^2-p^2}a_{n-2}=-\frac{1}{n\left(n \pm
 2p\right)}a_{n-2}$\\
 Discussion:

 The difference between the two values of $\sigma$ is
 $p-\left(-p\right)=2p$.

 (a) $2p$ not an integer: we get two independent
 Frobenius  series solutions.

 (b) $2p$ integer but $p$ not an integer: Assume
 p$>$0, Then for $\sigma=p$ we obtain a Frobenius solution, but
 for $\sigma=-p$ we find $a_{2p}=-\frac x{1}{2p\left(2p-2p\right)}a_{2p-2}$
 dividing by zero, so this second Frobenius solution does not
 exist.

 (c) $p$ an integer: We get only one series solution, which is
 now a power series.


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