91Ó°ÊÓ

Solve each differential equation by variation of parameters. $$y^{\prime \prime}-y=\cosh x$$

Solve the given system of differential equations by systematic elimination. $$\begin{aligned} &\frac{d x}{d t}+\frac{d y}{d t} \quad=e^{t}\\\ &-\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}+x+y=0 \end{aligned}$$

Solve the given system of differential equations by systematic elimination. $$\begin{aligned} &\frac{d x}{d t}=-x+z\\\ &\frac{d y}{d t}=-y+z\\\ &\frac{d z}{d t}=-x+y \end{aligned}$$

Find a linear differential operator that annihilates the given function. $$3+e^{x} \cos 2 x$$

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. $$4 y^{\prime \prime}-4 y^{\prime}+y=0 ; \quad e^{x / 2}, x e^{x / 2},(-\infty, \infty)$$

Discuss how the methods of undetermined coefficients and variation of parameters can be combined to solve the given differential equation. Carry out your ideas. $$y^{\prime \prime}-2 y^{\prime}+y=4 x^{2}-3+x^{-1} e^{x}$$

Solve the given differential equation by undetermined coefficients. $$2 y^{\prime \prime}-7 y^{\prime}+5 y=-29$$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$y=c_{1} e^{x}+c_{2} e^{5 x}$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+2 y^{\prime}+y=x^{2} e^{-x}$$