91Ó°ÊÓ

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

In Exercises 33-46 find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients in each solution. $$ 8 x^{2} y^{\prime \prime}+x\left(2+x^{2}\right) y^{\prime}+y=0 $$

Find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients. $$ x^{2} y^{\prime \prime}+x\left(1-3 x^{2}\right) y^{\prime}-4\left(1-3 x^{2}\right) y=0 $$

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. $$ y^{\prime \prime}+5 x y^{\prime}-\left(3-x^{2}\right) y=0, \quad y(0)=6, \quad y^{\prime}(0)=-2 $$

In Exercises 33-46 find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients in each solution. $$ 8 x^{2}\left(1-x^{2}\right) y^{\prime \prime}+2 x\left(1-13 x^{2}\right) y^{\prime}+\left(1-9 x^{2}\right) y=0 $$

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. $$ y^{\prime \prime}-2 x y^{\prime}-\left(2+3 x^{2}\right) y=0, \quad y(0)=2, \quad y^{\prime}(0)=-5 $$

$$ x^{2} y^{\prime \prime}+x\left(1-3 x^{2}\right) y^{\prime}-4\left(1-3 x^{2}\right) y=0 $$$$ x^{2}\left(1+x^{2}\right) y^{\prime \prime}+x\left(5+11 x^{2}\right) y^{\prime}+24 x^{2} y=0 $$

In Exercises 33-46 find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients in each solution. $$ x^{2}\left(1+x^{2}\right) y^{\prime \prime}-2 x\left(2-x^{2}\right) y^{\prime}+4 y=0 $$

Find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients. \(3 x^{2}\left(3+x^{2}\right) y^{\prime \prime}+x\left(3+11 x^{2}\right) y^{\prime}+\left(1+5 x^{2}\right) y=0\)

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. $$ y^{\prime \prime}-3 x y^{\prime}+\left(2+4 x^{2}\right) y=0, \quad y(0)=3, \quad y^{\prime}(0)=6 $$