91Ó°ÊÓ

Find the power series in \(x-x_{0}\) for the general solution. $$ y^{\prime \prime}-y=0 ; \quad x_{0}=3 $$

Do the following experiment for several choices of \(a_{0}\) and \(a_{1}\). (a) Use differential equations software to solve the initial value problem $$ (1-x) y^{\prime \prime}-(2-x) y^{\prime}+y=0, \quad y(0)=a_{0}, \quad y^{\prime}(0)=a_{1} $$ numerically on \((-r, r)\). (b) Find the coefficients \(a_{0}, a_{1}, \ldots, a_{N}\) in the power series solution \(y=\sum_{n=0}^{N} a_{n} x^{n}\) of (A), and graph $$ T_{N}(x)=\sum_{n=0}^{N} a_{n} x^{n} $$ and the solution obtained in (a) on \((-r, r)\). Continue increasing \(N\) until there's no perceptible difference between the two graphs. What happens as you let \(r \rightarrow 1 ?\)

Suppose \(y(x)=\sum_{n=0}^{\infty} a_{n}(x+1)^{n}\) on an open interval that contains \(x_{0}=-1 .\) Find a power series in \(x+1\) for $$ x y^{\prime \prime}+(4+2 x) y^{\prime}+(2+x) y . $$

Find the general solution of the given Euler equation on \((0, \infty)\). $$ 2 x^{2} y^{\prime \prime}+3 x y^{\prime}-y=0 $$

In Exercises 14-25 find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients in each solution. $$ 2 x^{2} y^{\prime \prime}+x(5+x) y^{\prime}-(2-3 x) y=0 $$

Find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients. $$ 25 x^{2} y^{\prime \prime}+x(15+x) y^{\prime}+(1+x) y=0 $$

Suppose \(y(x)=\sum_{n=0}^{\infty} a_{n}(x-2)^{n}\) on an open interval that contains \(x_{0}=2 .\) Find a power series in \(x-2\) for $$ x^{2} y^{\prime \prime}+2 x y^{\prime}-3 x y . $$

Find the power series in \(x-x_{0}\) for the general solution. $$ y^{\prime \prime}-(x-3) y^{\prime}-y=0 ; \quad x_{0}=3 $$

Find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients. $$ 2 x^{2}(2+x) y^{\prime \prime}+x^{2} y^{\prime}+(1-x) y=0 $$

Follow the directions of Exercise 16 for the initial value problem $$ (1+x) y^{\prime \prime}+3 y^{\prime}+32 y=0, \quad y(0)=a_{0}, \quad y^{\prime}(0)=a_{1} $$