91Ó°ÊÓ

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}+\frac{d y}{d t}=e^{t} \\ &-\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}+x+y=0 \end{aligned} $$

Solve the given differential equation. $$ x^{2} y^{\prime \prime}-7 x y^{\prime}+41 y=0 $$

In Problems 11-20, find the eigenvalues and eigenfunctions for the given boundary-value problem. $$ y^{\prime \prime}+\lambda y=0, y(0)=0, \quad y^{\prime}(\pi / 2)=0 $$

In Problems \(1-18\), solve each differential equation by variation of parameters. $$ y^{\prime \prime}-2 y^{\prime}+y=e^{t} \arctan t $$

The given two-parameter family is a solution of the indicated differential equation on the interval \((-\infty, \infty)\). Determine whether a member of the family can be found that satisfies the boundary conditions. \(y=c_{1} x^{2}+c_{2} x^{4}+3 ; x^{2} y^{\prime \prime}-5 x y^{\prime}+8 y=24\) (a) \(y(-1)=0, y(1)=4\) (b) \(y(0)=1, y(1)=2\) (c) \(y(0)=3, y(1)=0\) (d) \(y(1)=3, y(2)=15\)

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

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}-4 y=\left(x^{2}-3\right) \sin 2 x\)

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}+y=2 x \sin x\)

Solve the given differential equation. $$ x^{3} y^{\prime \prime \prime}-6 y=0 $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}+y=2 x \sin x $$