91Ó°ÊÓ

In Exercises 33-46 find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients in each solution. $$ x\left(1+x^{2}\right) y^{\prime \prime}+\left(4+7 x^{2}\right) y^{\prime}+8 x y=0 $$

Find a fundamental set of Frobenius solutions. Compute the terms involving \(x^{2 m+r_{1}}\), where \(0 \leq m \leq M(M\) at least 3\()\) and \(r_{1}\) is the root of the indicial equation. Optionally, write a computer program to implement the applicable recurrence formulas and take \(M>3\). \(x^{2}\left(1+x^{2}\right) y^{\prime \prime}+x\left(3+7 x^{2}\right) y^{\prime}+\left(1+8 x^{2}\right) y=0\)

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

In Exercises 33-46 find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients in each solution. $$ x^{2}\left(2+x^{2}\right) y^{\prime \prime}+x\left(3+x^{2}\right) y^{\prime}-y=0 $$

In Exercises 33-46 find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients in each solution. $$ 2 x^{2}\left(1+x^{2}\right) y^{\prime \prime}+x\left(3+8 x^{2}\right) y^{\prime}-\left(3-4 x^{2}\right) y=0 $$

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

In Exercises 33-46 find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients in each solution. $$ 9 x^{2} y^{\prime \prime}+3 x\left(3+x^{2}\right) y^{\prime}-\left(1-5 x^{2}\right) y=0 $$

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

Find the coefficients \(a_{0}, \ldots, a_{N}\) for \(N\) at least 7 in the series solution $$ y=\sum_{n=0}^{\infty} a_{n}\left(x-x_{0}\right)^{n} $$ of the initial value problem. Take \(x_{0}\) to be the point where the initial conditions are imposed. $$ \left(3+4 x+x^{2}\right) y^{\prime \prime}-\left(5+4 x-x^{2}\right) y^{\prime}-(2+x) y=0, \quad y(-2)=2, \quad y^{\prime}(-2)=-1 $$

Find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients. \(4 x^{2}(1+x) y^{\prime \prime}-4 x^{2} y^{\prime}+(1-5 x) y=0\)