91Ó°ÊÓ

Determine two linearly independent power series solutions to the given differential equation centered at \(x=0 .\) Also determine the radius of convergence of the series solutions. $$y^{\prime \prime}-y=0.$$

Use Rodrigues' formula to determine the Legendre polynomial of degree 3.

Determine whether \(x=0\) is an ordinary point or a regular singular point of the given differential equation. Then obtain two linearly independent solutions to the differential equation and state the maximum interval on which your solutions are valid. $$\left(1-x^{2}\right) y^{\prime \prime}-6 x y^{\prime}-4 y=0.$$

Determine two linearly independent power series solutions to the given differential equation centered at \(x=0 .\) Also determine the radius of convergence of the series solutions. $$y^{\prime \prime}-x^{2} y^{\prime}-2 x y=0.$$

Determine the radius of convergence of the given power series. $$\sum_{n=0}^{\infty} n ! x^{n}$$

(a) Given that \(\Gamma(1 / 2)=\sqrt{\pi}\) by Problem \(6,\) determine \(\Gamma(3 / 2)\) and \(\Gamma(-1 / 2)\) (b) Show that for positive integer \(n:\) $$ \Gamma\left(n+\frac{1}{2}\right)=\frac{(2 n) !}{2^{2 n} \cdot n !} \sqrt{\pi} $$ (c) Show that for positive integer \(n:\) $$ \Gamma\left(\frac{1}{2}-n\right)=\frac{(-1)^{n} \cdot 2^{2 n} \cdot n !}{(2 n) !} \sqrt{\pi} $$

Determine the roots of the indicial equation of the given differential equation. $$x^{2} y^{\prime \prime}-x(\cos x) y^{\prime}+5 e^{2 x} y=0$$

Determine the radius of convergence of the power series representation of the given function with center \(x_{0}\). $$f(x)=\frac{2 x}{x^{2}+16}, \quad x_{0}=1$$

Show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on \((0, \infty)\). $$4 x^{2} y^{\prime \prime}+3 x y^{\prime}+x y=0$$

Determine whether \(x=0\) is an ordinary point or a regular singular point of the given differential equation. Then obtain two linearly independent solutions to the differential equation and state the maximum interval on which your solutions are valid. $$x^{2} y^{\prime \prime}+\frac{3}{2} x y^{\prime}-\frac{1}{2}(1+x) y=0.$$