91Ó°ÊÓ

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 \((0, \infty)\). $$3 x^{2} y^{\prime \prime}-x(x+8) y^{\prime}+6 y=0$$

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 \((0, \infty)\). $$2 x^{2} y^{\prime \prime}-x(1+2 x) y^{\prime}+2(4 x-1) y=0$$

Consider the Chebyshev equation $$\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+a^{2} y=0, \quad (11.7.10)$$ where \(a\) is a constant. (a) Show that if \(a=N,\) a nonnegative integer, then Equation (11.7.10) has a polynomial solution of degree \(N .\) When suitably normalized, these polynomials are called the Chebyshev polynomials and are denoted by \(T_{N}(x).\) (b) Use Equation (11.7.10) to show that \(T_{N}(x)\) satisfies $$\left[\sqrt{1-x^{2}} T_{N}^{\prime}\right]^{\prime}+\frac{N}{\sqrt{1-x^{2}}} T_{N}=0.$$ (c) Use the result from (b) to prove that $$\int_{-1}^{1} \frac{T_{N}(x) T_{M}(x)}{\sqrt{1-x^{2}}} d x=0, \quad M \neq N.$$

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

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 \((0, \infty)\). $$3 x^{2} y^{\prime \prime}+x(7+3 x) y^{\prime}+(1+6 x) y=0$$

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 \(\left.\lambda_{k} \text { denote the positive zeros of } J_{0}(x) .\right]\)

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.