Chapter 11

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. $$y^{\prime \prime}+x y=0.$$

Determine all singular points of the given differential equation and classify them as regular or irregular singular points. $$y^{\prime \prime}+\frac{1}{1-x} y^{\prime}+x y=0$$

Use the relations \((11.6 .4)\) and \((11.6 .5)\) to show that if \(p\) is a half- integer, then Bessel's equation of order \(p\) has two linearly independent Frobenius series solutions.

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. $$y^{\prime \prime}-y=0.$$

Determine the radius of convergence of the given power series. $$\sum_{n=0}^{\infty} \frac{x^{n}}{2^{2 n}}$$

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. $$y^{\prime \prime}-x^{2} y=0.$$

Determine all singular points of the given differential equation and classify them as regular or irregular singular points. $$x^{2} y^{\prime \prime}+\frac{x}{x^{2}-4} y^{\prime}+\frac{1}{x(x-2)(x+2)^{2}} y=0$$

Determine the radius of convergence of the given power series. $$\sum_{n=0}^{\infty} \frac{(3 x)^{n}}{5^{3 n}}$$

Determine two linearly independent solutions to $$ x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\frac{9}{4}\right) y=0 $$ on the interval \((0, \infty)\).

Starting with \(P_{0}(x)=1\) and \(P_{1}(x)=x,\) use the recurrence relation \((n+1) P_{n+1}+n P_{n-1}=(2 n+1) x P_{n}, \quad n=1,2,3, \ldots\) to determine \(P_{2}, P_{3},\) and \(P_{4}\).

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. $$y^{\prime \prime}+2 x y^{\prime}+4 y=0.$$

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. $$\left(1-x^{2}\right) y^{\prime \prime}-6 x y^{\prime}-4 y=0.$$

Determine all singular points of the given differential equation and classify them as regular or irregular singular points. $$x^{2} y^{\prime \prime}+\frac{x}{\left(1-x^{2}\right)^{2}} y^{\prime}+y=0$$

Verify that \(y(x)=x J_{1}(x)\) is a solution to the differential equation $$ x y^{\prime \prime}-y^{\prime}+x y=0, \quad x>0 $$

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

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. $$y^{\prime \prime}-2 x y^{\prime}-2 y=0.$$

Determine the radius of convergence of the given power series. $$\sum_{n=0}^{\infty} \frac{x^{n}}{n^{2}}$$

Determine the roots of the indicial equation of the given differential equation. Also obtain the general form of two linearly independent solutions to the differential equation on an interval \((0, R) .\) Finally, if \(r_{1}-r_{2}\) equals a positive integer, obtain the recurrence relation and determine whether the constant \(A\) in $$y_{2}(x)=A y_{1}(x) \ln x+x^{r_{2}} \sum_{n=0}^{\infty} b_{n} x^{n}$$ is zero or nonzero. $$x^{2} y^{\prime \prime}+x^{2} y^{\prime}-(2+x) y=0$$

Let \(\Gamma(p)\) denote the gamma function. Show that $$\Gamma(p+1)[(p+1)(p+2) \cdots(p+k)]=\Gamma(p+k+1)$$

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. $$\sum_{n=0}^{\infty} \frac{2^{n} x^{n}}{n}$$

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. $$x y^{\prime \prime}+y^{\prime}+2 y=0.$$

Determine all singular points of the given differential equation and classify them as regular or irregular singular points. $$(x-2)^{2} y^{\prime \prime}+(x-2) e^{x} y^{\prime}+\frac{4}{x} y=0$$

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. $$x y^{\prime \prime}+2 y^{\prime}+x y=0.$$

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. $$y^{\prime \prime}+x y=0.$$

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

Determine all singular points of the given differential equation and classify them as regular or irregular singular points. $$y^{\prime \prime}+\frac{2}{x(x-3)} y^{\prime}-\frac{1}{x^{3}(x+3)} y=0$$

For \(p>0\) and \(a>0\) express the following integral in terms of the gamma function: $$ \int_{0}^{\infty} t^{p-1} e^{-a t} d t $$

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. $$2 x y^{\prime \prime}+5(1-2 x) y^{\prime}-5 y=0.$$

Let \(Q(x)\) be a polynomial of degree less than \(N .\) Prove that \(\int_{-1}^{1} Q(x) P_{N}(x) d x=0\).

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. $$y^{\prime \prime}+x y^{\prime}+3 y=0.$$

Determine the roots of the indicial equation of the given differential equation. $$x^{2} y^{\prime \prime}+x(1-x) y^{\prime}-7 y=0$$

(a) By making the change of variables \(t=x^{2}\) in the integral that defines the gamma function, show that $$ \Gamma(1 / 2)=2 \int_{0}^{\infty} e^{-x^{2}} d x $$ (b) Use your result from (a) to show that $$ [\Gamma(1 / 2)]^{2}=4 \int_{0}^{\infty} \int_{0}^{\infty} e^{-\left(x^{2}+y^{2}\right)} d x d y $$ (c) By changing to polar coordinates, evaluate the double integral in (b) and hence show that $$ \Gamma(1 / 2)=\sqrt{\pi} $$

Determine the radius of convergence of the given power series. $$\sum_{n=0}^{\infty} \frac{5^{n} x^{n}}{n !}$$

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. $$x y^{\prime \prime}+y^{\prime}+x y=0.$$

Determine the roots of the indicial equation of the given differential equation. $$4 x^{2} y^{\prime \prime}+x e^{x} y^{\prime}-y=0$$

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. $$y^{\prime \prime}-x^{2} y^{\prime}-3 x y=0.$$

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

(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 \(n:\) $$ \Gamma\left(\frac{1}{2}-n\right)=\frac{(-1)^{n} \cdot 2^{2 n} \cdot n !}{(2 n) !} \sqrt{\pi} $$

Show that $$\frac{d^{2} Y}{d \phi^{2}}+\cot \phi \frac{d Y}{d \phi}+\alpha(\alpha+1) Y=0, \quad 0<\phi<\pi,$$ is transformed into Legendre's equation by the change of variables \(x=\cos \phi\).

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. $$\left(1+4 x^{2}\right) y^{\prime \prime}-8 y=0.$$

Determine the roots of the indicial equation of the given differential equation. $$4 x y^{\prime \prime}-x y^{\prime}+2 y=0$$

Let \(J_{p}(x)\) denote the Bessel function of the first kind of order \(p .\) Show that $$ \frac{d}{d x}\left(x^{-p} J_{p}(x)\right)=-x^{-p} J_{p+1}(x) $$

Deal with Hermite's equation: $$y^{\prime \prime}-2 x y^{\prime}+2 \alpha y=0, \quad-\infty

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. $$y^{\prime \prime}+2 x^{2} y^{\prime}+2 x y=0.$$

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

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. $$x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\frac{1}{4}\right) y=0.$$

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 all values of the constant \(\alpha\) for which $$x^{2} y^{\prime \prime}+x(1-2 x) y^{\prime}+\left[2(\alpha-1) x-\alpha^{2}\right] y=0$$ has two linearly independent Frobenius series solutions on \((0, \infty)\)

Deal with Hermite's equation: $$y^{\prime \prime}-2 x y^{\prime}+2 \alpha y=0, \quad-\infty