Chapter 11

Determine two linearly independent solutions to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}+x y^{\prime}-(1+x) y=0$$

Determine a series solution to the initial-value problem $$\left(1+2 x^{2}\right) y^{\prime \prime}+7 x y^{\prime}+2 y=0$$ $$y(0)=0, \quad y^{\prime}(0)=1$$

Consider the general "perturbed" Cauchy-Euler equation $$\begin{aligned} &x^{2} y^{\prime \prime}+x[1-(a+b)+c x] y^{\prime}+(a b+d x) y=0, x>0, \quad (11.7.12) \end{aligned}$$ where \(a, b, c, d\) are constants. Assuming that \(a\) and \(b\) are distinct and do not differ by an integer, determine two linearly independent Frobenius series solutions to Equation (11.7.12). [Hint: Use symmetry to get the second solution.]

Consider the differential equation $$x^{2} y^{\prime \prime}+x y^{\prime}+(1-x) y=0, \quad x>0$$ (a) Find the indicial equation, and show that the roots are \(r=\pm i\) (b) Determine the first three terms in a complex valued Frobenius series solution to Equation \((11.4 .15)\) (c) Use the solution in (b) to determine two linearly independent real-valued solutions to Equation (11.4.15).

Determine the Fourier-Bessel expansion in the functions \(J_{0}\left(\lambda_{k} x\right)\) of \(f(x)=x^{2}\) on the interval (0,1) [Here the \(\left.\lambda_{k} \text { denote the positive zeros of } J_{0}(x) .\right]\)

Consider the differential equation $$x^{2} y^{\prime \prime}+x(1+b x) y^{\prime}+\left[b(1-N) x-N^{2}\right] y=0, x>0, \quad \quad (11.7.13)$$ where \(N\) is a positive integer and \(b\) is a constant. (a) Show that the roots of the indicial equation are \(r=\pm N.\) (b) Show that the Frobenius series solution corresponding to \(r=N\) is $$y_{1}(x)=a_{0} x^{N} \sum_{n=0}^{\infty} \frac{(2 N) !(-b)^{n}}{(2 N+n) !} x^{n}$$ and that by an appropriate choice of \(a_{0},\) one solution to ( 11.7 .13 ) is $$y_{1}(x)=x^{-N}\left[e^{-b x}-\sum_{n=0}^{2 N-1} \frac{(-b x)^{n}}{n !}\right].$$ (c) Show that Equation (11.7.13) has a second linearly independent Frobenius series solution that can be taken as $$y_{2}(x)=x^{-N} \sum_{n=0}^{2 N-1} \frac{(-b x)^{n}}{n !}.$$ Hence, conclude that Equation (11.7.13) has linearly independent solutions $$y_{1}(x)=x^{-N} e^{-b x}, y_{2}(x)=x^{-N} \sum_{n=0}^{2 N-1} \frac{(-b x)^{n}}{n !}.$$

Determine two linearly independent solutions to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}-x(x+3) y^{\prime}+4 y=0$$

(a) Determine a series solution to the initial-value problem $$4 y^{\prime \prime}+x y^{\prime}+4 y=0, \quad y(0)=1, \quad y^{\prime}(0)=0$$ (b) Find a polynomial that approximates the solution to Equation (11.2.19) with an error less than \(10^{-5}\) on the interval [-1,1] [Hint: The series obtained is a convergent alternating series.

Determine the first five nonzero terms in each of two linearly independent Frobenius series solutions to $$ 3 x^{2} y^{\prime \prime}+x\left(1+3 x^{2}\right) y^{\prime}-2 x y=0, \quad x>0 $$

Let \(J_{p}(x)\) denote the Bessel function of the first kind of order \(p,\) and let \(\lambda\) be a positive real number. If \(u(x)=\) \(J_{p}(\lambda x),\) show that \(u\) satisfies the differential equation $$ \frac{d^{2} u}{d x^{2}}+\frac{1}{x} \frac{d u}{d x}+\left(\lambda^{2}-\frac{p^{2}}{x^{2}}\right) u=0 $$

Show that the change of variables \(y=x^{1 / 2} u\) transforms the differential equation $$y^{\prime \prime}+\left(1-\frac{3}{4 x^{2}}\right) y=0$$ into the Bessel equation $$x^{2} u^{\prime \prime}+x u^{\prime}+\left(x^{2}-1\right) u=0$$ and thereby write the general solution to the given differential equation.

Consider the differential equation $$ 4 x^{2} y^{\prime \prime}-4 x^{2} y^{\prime}+(1+2 x) y=0 $$ (a) Show that the indicial equation has only one root, and find the corresponding Frobenius series solution. (b) Use the reduction of order technique to find a second linearly independent solution on \((0, \infty)\) [Hint: To evaluate \(\int \frac{e^{x}}{x} d x,\) expand \(e^{x}\) in a Maclaurin series. \(]^{10}\)

Let \(\lambda\) and \(\mu\) be positive real numbers. Then \(J_{p}(\lambda x)\) and \(J_{p}(\mu x)\) satisfy $$\frac{d}{d x}\left\\{x \frac{d}{d x}\left[J_{p}(\lambda x)\right]\right\\}+\left(\lambda^{2} x-\frac{p^{2}}{x}\right) J_{p}(\lambda x)=0$$ $$\frac{d}{d x}\left\\{x \frac{d}{d x}\left[J_{p}(\mu x)\right]\right\\}+\left(\mu^{2} x-\frac{p^{2}}{x}\right) J_{p}(\mu x)=0$$ respectively. (a) Show that for \(\lambda \neq \mu\) $$\int_{0}^{1} x J_{p}(\lambda x) J_{p}(\mu x) d x$$ $$=\frac{\mu J_{p}(\lambda) J_{p}^{\prime}(\mu)-\lambda J_{p}(\mu) J_{p}^{\prime}(\lambda)}{\lambda^{2}-\mu^{2}}$$ [Hint: Multiply (1 1.6.37) by \(J_{p}(\mu x),(11.6 .38)\) by \(J_{p}(\lambda x),\) subtract the resulting equations and integrate over \((0,1) .\) If \(\lambda\) and \(\mu\) are distinct zeros of \(J_{p}(x),\) what does your result imply? (b) In order to compute \(\int_{0}^{1} x\left[J_{p}(\mu x)\right]^{2} d x,\) we take the limit as \(\lambda \rightarrow \mu\) in \((11.6 .39) .\) Use L'Hopital's rule to compute this limit and thereby show that $$\begin{array}{rl}\int_{0}^{1} & x\left[J_{p}(\mu x)\right]^{2} d x \\ & =\frac{\mu\left[J_{p}^{\prime}(\mu)\right]^{2}-J_{p}(\mu) J_{p}^{\prime}(\mu)-\mu J_{p}(\mu) J_{p}^{\prime \prime}(\mu)}{2 \mu}\end{array}$$ Substituting from Bessel's equation for \(J_{p}^{\prime \prime}(\mu)\) show that \((11.6 .40)\) can be written as $$ \begin{array}{l} \int_{0}^{1} x\left[J_{p}(\mu x)\right]^{2} d x \\ \quad=\frac{1}{2}\left\\{\left[J_{p}^{\prime}(\mu)\right]^{2}+\left(1-\frac{p^{2}}{\mu^{2}}\right)\left[J_{p}(\mu)\right]^{2}\right\\} \end{array} $$ (c) In the case when \(\mu\) is a zero of \(J_{p}(x),\) use \((11.6 .26)\) to show that your result in (b) can be written as $$ \int_{0}^{1} x\left[J_{p}(\mu x)\right]^{2} d x=\frac{1}{2}\left[J_{p+1}(\mu)\right]^{2} $$

Determine two linearly independent solutions to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}-x^{2} y^{\prime}-2 y=0$$

Find two linearly independent solutions to $$ x^{2} y^{\prime \prime}+x(3-2 x) y^{\prime}+(1-2 x) y=0 $$ on \((0, \infty)\)

Determine two linearly independent solutions to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}-x^{2} y^{\prime}-(3 x+2) y=0$$

Determine two linearly independent solutions to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}+x(5-x) y^{\prime}+4 y=0$$

Determine two linearly independent solutions to the given differential equation on \((0, \infty)\) $$4 x^{2} y^{\prime \prime}+4 x(1-x) y^{\prime}+(2 x-9) y=0$$

Determine two linearly independent solutions to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}+2 x(2+x) y^{\prime}+2(1+x) y=0$$

Determine two linearly independent solutions to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}-x(1-x) y^{\prime}+(1-x) y=0$$

Determine two linearly independent solutions to the given differential equation on \((0, \infty)\) $$4 x^{2} y^{\prime \prime}+4 x(1+2 x) y^{\prime}+(4 x-1) y=0$$

Determine two linearly independent solutions to the given differential equation on \((0, \infty)\) $$4 x^{2} y^{\prime \prime}-(3+4 x) y=0$$

Determine a Frobenius series solution to the given differential equation and use the reduction of order technique to find a second linearly independent solution on \((0, \infty)\) $$x y^{\prime \prime}-x y^{\prime}+y=0$$

Determine a Frobenius series solution to the given differential equation and use the reduction of order technique to find a second linearly independent solution on \((0, \infty)\) $$x^{2} y^{\prime \prime}+x(4+x) y^{\prime}+(2+x) y=0$$

Consider the Laguerre differential equation $$x^{2} y^{\prime \prime}+x(1-x) y^{\prime}+N x y=0$$ where \(N\) is a constant. Show that in the case when \(N\) is a positive integer, Equation \((11.5 .40)\) has a solution that is a polynomial of degree \(N,\) and find it. When properly normalized, these solutions are called the Laguerre polynomials.

Consider the differential equation $$x^{2} y^{\prime \prime}+x(1+2 N+x) y^{\prime}+N^{2} y=0, \quad(11.5 .41)$$ where \(N\) is a positive integer. (a) Show that there is only one Frobenius series solution and that it terminates after \(N+1\) terms. Find this solution. (b) Show that the change of variables \(Y=x^{N} y\) transforms Equation \((11.5 .41)\) into the Laguerre differential equation (11.5.40).