91Ó°ÊÓ

In Exercises 19-24 find the power series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) for the function \(f(x)\). $$ f(x)=\left(1+x^{2}\right) \sum_{n=1}^{\infty} \frac{x^{n}}{n} $$

Find a fundamental set of solutions for the differential equations in Exercises 13-22. $$ x^{2} y^{\prime \prime}+x(x-3) y^{\prime}+4 y=0 $$

In Exercises 19-22, find the general solution. Then find the solution that satisfies the given initial conditions. $$ (x-3)^{2} y^{\prime \prime}+5(x-3) y^{\prime}+4 y=0, y(4)=1 \text { and } y^{\prime}(4)=1 $$

In Exercises 21-24, verify that \(x_{0}=0\) is an ordinary point. Find \(S_{4}\), the partial sum of order 4 for two linearly independent solutions. Estimate the radius of convergence of the solutions. $$ (1+x) y^{\prime \prime}+y=0 $$

In Exercises 19-24 find the power series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) for the function \(f(x)\). $$ f(x)=\sum_{n=0}^{\infty} x^{n}-\sum_{n=1}^{\infty} \frac{x^{n-1}}{n} $$

In Exercises 11-25, find two Frobenius series solutions. $$ 2 x y^{\prime \prime}+y^{\prime}+y=0 $$

In Exercises 19-22, find the general solution. Then find the solution that satisfies the given initial conditions. $$ (x-1)^{2} y^{\prime \prime}-6 y=0, y(0)=1 \text { and } y^{\prime}(0)=1 \text {. } $$

In Exercises 19-24 find the power series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) for the function \(f(x)\). $$ f(x)=\left(1+x^{2}\right) \sin x $$

In Exercises 11-25, find two Frobenius series solutions. $$ 2 x^{2} y^{\prime \prime}-x y^{\prime}+(1+x) y=0 $$

In Exercises 21-24, verify that \(x_{0}=0\) is an ordinary point. Find \(S_{4}\), the partial sum of order 4 for two linearly independent solutions. Estimate the radius of convergence of the solutions. $$ y^{\prime \prime}-(\cos x) y=0 $$