91Ó°ÊÓ

Find a power series solution for the following differential equations. $$ (x-7) y^{\prime}+2 y=0 $$

Find a power series solution for the following differential equations. $$ \left(1+x^{2}\right) y^{\prime \prime}-4 x y^{\prime}+6 y=0 $$

Find a power series solution for the following differential equations. $$ x^{2} y^{\prime \prime}-x y^{\prime}-3 y=0 $$

Find a power series solution for the following differential equations. $$ y^{\prime \prime}-8 y^{\prime}=0, \quad y(0)=-2, \quad y^{\prime}(0)=10 $$

Find a power series solution for the following differential equations. $$ y^{\prime \prime}-2 x y=0, \quad y(0)=1, \quad y^{\prime}(0)=-3 $$

Find a power series solution for the following differential equations. \(\begin{array}{llll}116 . & \text { The } & \text { differential } & \text { equation }\end{array}\) \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-1\right) y=0\) is a Bessel equation of order 1. Use a power series of the form \(y=\sum_{n=0}^{\infty} a_{n} x^{n}\) to find the solution.

True or False? Justify your answer with a proof or a counterexample. If \(y\) and \(z\) are both solutions to \(y^{n}+2 y^{\prime}+y=0\), then \(y+z\) is also a solution.

True or False? Justify your answer with a proof or a counterexample. The following system of algebraic equations has a unique solution: $$ \begin{array}{l} 6 z_{1}+3 z_{2}=8 \\ 4 z_{1}+2 z_{2}=4 \end{array} $$

True or False? Justify your answer with a proof or a counterexample. \(y=e^{x} \cos (3 x)+e^{x} \sin (2 x)\) is a solution to the second-order differential equation \(y^{\prime \prime}+2 y^{\prime}+10=0\).

True or False? Justify your answer with a proof or a counterexample. To find the particular solution to a second-order differential equation, you need one initial condition.