91Ó°ÊÓ

Show that the indicial roots differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about the regular singular point \(x_{0}=0\). Form the general solution on \((0, \infty)\). $$ x y^{\prime \prime}+y=0 $$

Show that the indicial roots differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about the regular singular point \(x_{0}=0\). Form the general solution on \((0, \infty)\). $$ x y^{\prime \prime}+y^{\prime}+y=0 $$

Show that the indicial roots differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about the regular singular point \(x_{0}=0\). Form the general solution on \((0, \infty)\). $$ x y^{*}-x y^{*}+y=0 $$

Show that \(i^{-v} J_{,}(i x), i^{2}=-1\) is a real function. The function defined by \(I_{v}(x)=i^{-x} J_{v}(i x)\) is called a modified Bessel function of the first kind of order \(\nu\).

Use the change of variables \(s=\frac{2}{\alpha} \sqrt{\frac{k}{m}} e^{-m / 2}\) to show that the differential equation of an aging spring \(m x^{\prime \prime}+k e^{-a x} x=0, \alpha>0\), becomes $$ s^{2} \frac{d^{2} x}{d s^{2}}+s \frac{d x}{d s}+s^{2} x=0 $$

Show that the equation $$ \sin \theta \frac{d^{2} y}{d \theta^{2}}+\cos \theta \frac{d y}{d \theta}+n(n+1)(\sin \theta) y=0 $$ can be transformed in Legendre's equation by means of the substitution \(x=\cos \theta\).

The Legendre polynomials are also generated by Rodrigues' formula $$ P_{n}(x)=\frac{1}{2^{2} n !} \frac{d^{n}}{d x^{n}}\left(x^{2}-1\right)^{n} . $$ Verify the results for \(n=0,1,2,3\).