91Ó°ÊÓ

Show that \(J_{0}(k x)\), where \(k\) is a constant, satisfies the differential equation $$ x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}+k^{2} x y=0 $$

Find power series solutions in powers of \(x\) of each of the differential equations in Exercises \(1-10 .\) $$ \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=0 $$

Locate and classify the singular points of each of the differential equations in Exercises \(1-4\) $$ \left(x^{2}-3 x\right) \frac{d^{2} y}{d x^{2}}+(x+2) \frac{d y}{d x}+y=0 $$

Find power series solutions in powers of \(x\) of each of the differential equations in Exercises \(1-10 .\) $$ \frac{d^{2} y}{d x^{2}}+8 x \frac{d y}{d x}-4 y=0 $$

Locate and classify the singular points of each of the differential equations in Exercises \(1-4\) $$ \left(x^{3}+x^{2}\right) \frac{d^{2} y}{d x^{2}}+\left(x^{2}-2 x\right) \frac{d y}{d x}+4 y=0 $$

Show that the transformation $$ y=\frac{u(x)}{\sqrt{x}} $$ reduces the Bessel equation of order \(p\), Equation \((6.101)\), to the form $$ \frac{d^{2} u}{d x^{2}}+\left[1+\left(\frac{1}{4}-p^{2}\right) \frac{1}{x^{2}}\right] u=0 . $$

Locate and classify the singular points of each of the differential equations in Exercises \(1-4\) $$ \left(x^{4}-2 x^{3}+x^{2}\right) \frac{d^{2} y}{d x^{2}}+2(x-1) \frac{d y}{d x}+x^{2} y=0 $$

Find power series solutions in powers of \(x\) of each of the differential equations in Exercises \(1-10 .\) $$ \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+\left(2 x^{2}+1\right) y=0 $$

Locate and classify the singular points of each of the differential equations in Exercises \(1-4\) $$ \left(x^{5}+x^{4}-6 x^{3}\right) \frac{d^{2} y}{d x^{2}}+x^{2} \frac{d y}{d x}+(x-2) y=0 $$

Find power series solutions in powers of \(x\) of each of the differential equations in Exercises \(1-10 .\) $$ \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+\left(x^{2}-4\right) y=0 $$