91Ó°ÊÓ

In Exercises 1-14, solve the given equation exactly using a technique from a previous chapter. Then find a power series solution and verify that it is the series expansion of the exact solution. $$ y^{\prime \prime}+y=0 $$

In Exercises 10-17, find the general solution to each example of Euler's equation. $$ -x^{2} y^{\prime \prime}+x y^{\prime}-y=0 $$

In Exercises 1-14, solve the given equation exactly using a technique from a previous chapter. Then find a power series solution and verify that it is the series expansion of the exact solution. $$ y^{\prime \prime}+4 y=0 $$

In Exercises 10-17, find the general solution to each example of Euler's equation. $$ x^{2} y^{\prime \prime}-2 x y^{\prime}-4 y=0 $$

Find the Taylor series about the given point for each of the functions in Exercises 13-18. In each case find the radius of convergence. $$ f(x)=\sin x^{2} \text { about } x_{0}=0 $$

In Exercises 11-25, find two Frobenius series solutions. $$ 4 x^{2} y^{\prime \prime}+4 x y^{\prime}-\left(4 x^{2}+1\right) y=0 $$

In Exercises 11-25, find two Frobenius series solutions. $$ x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-1 / 16\right) y=0 $$

In Exercises 10-17, find the general solution to each example of Euler's equation. $$ x^{2} y^{\prime \prime}+9 x y^{\prime}+16 y=0 $$

Find the Taylor series about the given point for each of the functions in Exercises 13-18. In each case find the radius of convergence. $$ f(x)=\sin x \text { about } x_{0}=\pi $$

In Exercises 15-20, verify that \(x_{0}=0\) is an ordinary point of the given differential equation. Then find two linearly independent solutions to the differential equation valid near \(x_{0}=0\). Estimate the radius of convergence of the solutions. $$ y^{\prime \prime}+x^{2} y=0 $$