91Ó°ÊÓ

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

The differential equation \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-1\right) y=0\) 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 \(y^{\prime \prime}+2 y^{\prime}+y=0\) 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: \(6 z_{1}+3 z_{2}=8\) \(4 z_{1}+2 z_{2}=4\)

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation. $$y^{\prime \prime}-2 y=0$$

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation. $$y^{\prime \prime}-3 y+2 y=\cos (t)$$

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation. $$\left(\frac{d y}{d t}\right)^{2}+y y^{\prime}=1$$

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation. $$\frac{d^{2} y}{d t^{2}}+t \frac{d y}{d t}+\sin ^{2}(t) y=e^{t}$$

For the following problems, find the general solution. $$y^{\prime \prime}+9 y=0$$

For the following problems, find the general solution. $$y^{\prime \prime}+2 y^{\prime}+y=0$$