Question

\( \rightarrow \) Find Whe sourin of Revcci founa \[ x^{2} \frac{d^{2} y}{d x}+x \frac{d y}{d x}+\left(x^{2}-p^{2}\right) y=0 \] bole (17) As (giren) \[ \begin{array}{l} \Rightarrow x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-y^{2}\right) y=0-\infty \\ \because x\left(x y^{\prime}\right)^{\prime}=x\left[y^{\prime}+x y^{\prime \prime}\right]=x y^{\prime}+x^{\prime} y^{\prime \prime} \end{array} \] Hence: \[ \text { (1) } \Rightarrow x\left(x y^{\prime}\right)^{\prime}+\left(x^{2}-p^{2}\right) y=0 \] Let. \[ \begin{array}{l} y=\sum_{n=0}^{\infty} a_{n} x^{n+s} \\ \Rightarrow y^{\prime}=\sum_{n \rightarrow 0}(n+s) a_{n} x^{n+s-1} \\ \Rightarrow\left(x y^{\prime}\right)=\sum_{n=0}^{\infty}(n+s) a_{n} x^{n+s} \text {. } \\ \Rightarrow\left(x y^{\prime}\right)^{\prime}=\sum_{n=0}(n+s)^{2} a_{n} x^{n+s-1} \\ =x\left(n y^{\prime}\right)^{\prime}=\sum_{n=0}^{n}(n+s)^{2} a_{n} x^{n+s} \text {. } \\ \text { 4. } x^{2} y=\sum_{n=0}^{m} a_{n} x^{n+s+2} \\ \text { (4) } \rightarrow\left(-p^{2} y=-\frac{1}{n=0} a_{n} x^{n+5} \cdot p^{2}\right. \\ \end{array} \] Table: Get:

          \( \rightarrow \) Find Whe sourin of Revcci founa
\[
x^{2} \frac{d^{2} y}{d x}+x \frac{d y}{d x}+\left(x^{2}-p^{2}\right) y=0
\]
bole
(17) As (giren)
\[
\begin{array}{l}
\Rightarrow x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-y^{2}\right) y=0-\infty \\
\because x\left(x y^{\prime}\right)^{\prime}=x\left[y^{\prime}+x y^{\prime \prime}\right]=x y^{\prime}+x^{\prime} y^{\prime \prime}
\end{array}
\]

Hence:
\[
\text { (1) } \Rightarrow x\left(x y^{\prime}\right)^{\prime}+\left(x^{2}-p^{2}\right) y=0
\]

Let.
\[
\begin{array}{l}
y=\sum_{n=0}^{\infty} a_{n} x^{n+s} \\
\Rightarrow y^{\prime}=\sum_{n \rightarrow 0}(n+s) a_{n} x^{n+s-1} \\
\Rightarrow\left(x y^{\prime}\right)=\sum_{n=0}^{\infty}(n+s) a_{n} x^{n+s} \text {. } \\
\Rightarrow\left(x y^{\prime}\right)^{\prime}=\sum_{n=0}(n+s)^{2} a_{n} x^{n+s-1} \\
=x\left(n y^{\prime}\right)^{\prime}=\sum_{n=0}^{n}(n+s)^{2} a_{n} x^{n+s} \text {. } \\
\text { 4. } x^{2} y=\sum_{n=0}^{m} a_{n} x^{n+s+2} \\
\text { (4) } \rightarrow\left(-p^{2} y=-\frac{1}{n=0} a_{n} x^{n+5} \cdot p^{2}\right. \\
\end{array}
\]

Table:

Get:

→ Find Whe sourin of Revcci founa

x^2(d^2 y)/(d x)+x (d y)/(d x)+(x^2-p^2) y=0

bole
(17) As (giren)

⇒ x^2 y^''+x y^'+(x^2-y^2) y=0-∞
∵ x(x y^')^'=x[y^'+x y^'']=x y^'+x^' y^''

Hence:

(1) ⇒ x(x y^')^'+(x^2-p^2) y=0

Let.

y=∑n=0^∞ an x^n+s
⇒ y^'=∑n → 0(n+s) an x^n+s-1
⇒(x y^')=∑n=0^∞(n+s) an x^n+s.
⇒(x y^')^'=∑n=0(n+s)^2 an x^n+s-1

=x(n y^')^'=∑n=0^n(n+s)^2 an x^n+s.
4. x^2 y=∑n=0^m an x^n+s+2
(4) →(-p^2 y=-(1)/(n=0) an x^n+5· p^2.

Table:

Get:

Added by Alfonso B.

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