91Ó°ÊÓ

Determine whether the points \(x=0, x=1\) and \(x=2\) are ordinary points of the differential equation $$ (x-1) y^{\prime \prime}+x y^{\prime}+(1 / x) y=0 $$ and discuss the solutions of this equation around these points.

Find a power series solution of the differential equation $$ y^{\prime}-y=0 $$

Find the power series solution of the differential equation $$ y^{\prime \prime}+x y^{\prime}+\left(x^{2}+2\right) y=0 $$ in powers of \(x\) (that is, about \(x_{0}=0\) ).

Find a series solution of the differential equation $$ x^{2} y^{\prime \prime}+y^{\prime}+y=0 $$

Find a series solution in power of \(x\) of the differential Equation $$ x^{2} y^{\prime \prime}+4 x y^{\prime}+2 y=0 $$

Write a power series expansion for the function $$ y=(1+x)^{p} $$ where \(p\) is an arbitrary constant, by finding the differential equation that it satisfies, and solving that equation by the power series method.

Find a series solution for the equation $$ y^{\prime \prime}=x y^{2}-y^{\prime} $$ having initial values $$ y(0)=2, y^{\prime}(0)=1 . $$

Find, in a series expansion, the general solution of the Airy equation, $$ x^{\prime \prime}-s x=0, \quad(-\infty

Solve the differential equation $$ \begin{aligned} &\left(d^{2} y / d x^{2}\right)+y=0, \text { such that } \\ &y(0)=0, y^{\prime}(0)=1, \text { by the } \end{aligned} $$ power series method.

Find a sequence of functions \(\left\\{y_{n}\right\\}\) which converge to the solution of the initial value problem $$ \begin{aligned} &(d y / d x)=x^{2} \\ &\text { with } y(2)=1 \end{aligned} $$