91Ó°ÊÓ

Find the general solution of the given differential equation on \((0, \infty)\). $$ 4 x^{2} y^{\prime \prime}+4 x y^{\prime}+\left(4 x^{2}-25\right) y=0 $$

Find two linearly independent power series solutions for each differential equation about the ordinary point \(x=0\). $$y^{\prime \prime}-x y^{\prime}+2 y=0$$

Determine the singular points of each differential equation. Classify each singular point as regular or irregular. $$ y^{*}-\frac{1}{x} y^{*}+\frac{1}{(x-1)^{3}} y=0 $$

Find the interval of convergence of the given power series. $$ \sum_{k=0}^{\infty} \frac{5^{k}}{k !} x^{k} $$

In Problems 3-6 specify the regular and irregular singular points of the given differential equation. $$ \left(x^{3}-10 x^{2}+25 x\right) y^{\prime \prime}+y=0 $$

Find the general solution of the given differential equation on \((0, \infty)\). $$ 16 x^{2} y^{\prime \prime}+16 x y^{\prime}+\left(16 x^{2}-1\right) y=0 $$

Find the general solution of the given differential equation on \((0, \infty)\). $$ x y^{\prime \prime}+y^{\prime}+x y=0 $$

Determine the singular points of each differential equation. Classify each singular point as regular or irregular. $$ \left(x^{3}+4 x\right) y^{\prime \prime}-2 x y^{\prime}+6 y=0 $$

Find two linearly independent power series solutions for each differential equation about the ordinary point \(x=0\). $$y^{\prime \prime}+x^{2} y^{\prime}+x y=0$$

In Problems 3-6 specify the regular and irregular singular points of the given differential equation. $$ x^{2}\left(x^{2}-9\right)^{2} y^{\prime \prime}-\left(x^{2}-9\right) y^{\prime}+x y=0 $$