91Ó°ÊÓ

Find the coefficients \(a_{0}, \ldots, a_{N}\) for \(N\) at least 7 in the series solution \(y=\sum_{n=0}^{\infty} a_{n} x^{n}\) of the initial value problem. $$ x^{2} y^{\prime \prime}-(6-7 x) y^{\prime}+8 y=0, \quad y(1)=1, \quad y^{\prime}(1)=-2 $$

Find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients. $$ 4 x^{2} y^{\prime \prime}+(1+4 x) y=0 $$

Find \(a_{0}, \ldots, a_{N}\) for \(N\) at least 7 in the power series solution \(y=\sum_{n=0}^{\infty} a_{n} x^{n}\) of the initial value problem. $$ \left(1+2 x^{2}\right) y^{\prime \prime}-9 x y^{\prime}-6 y=0, \quad y(0)=1, \quad y^{\prime}(0)=-1 $$

Find the general solution of the given Euler equation on \((0, \infty)\). $$ x^{2} y^{\prime \prime}+3 x y^{\prime}+5 y=0 $$

Find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients. $$ 36 x^{2}(1-2 x) y^{\prime \prime}+24 x(1-9 x) y^{\prime}+(1-70 x) y=0 $$

Find \(a_{0}, \ldots, a_{N}\) for \(N\) at least 7 in the power series solution \(y=\sum_{n=0}^{\infty} a_{n} x^{n}\) of the initial value problem. $$ \left(1+8 x^{2}\right) y^{\prime \prime}+2 y=0, \quad y(0)=2, \quad y^{\prime}(0)=-1 $$

Find the general solution of the given Euler equation on \((0, \infty)\). $$ 9 x^{2} y^{\prime \prime}+15 x y^{\prime}+y=0 $$

Do the following experiment for various choices of real numbers \(a_{0}, a_{1},\) and \(r,\) with \(0

In Exercises \(11-15\) find a power series solution \(y(x)=\sum_{n=0}^{\infty} a_{n} x^{n} .\) $$ \left(1+2 x^{2}\right) y^{\prime \prime}+(2-3 x) y^{\prime}+4 y $$

Do the next experiment for various choices of real numbers \(a_{0}, a_{1},\) and \(r,\) with \(0