91Ó°ÊÓ

Solve the given differential equation. \(x^{2} y^{\prime \prime}-x y^{\prime}+2 y=0\)

In Problems 1-36 find the general solution of the given differential equation. \(12 y^{\prime \prime}-5 y^{\prime}-2 y=0\)

Given that \(x=c_{1} \cos \omega t+c_{2} \sin \omega t\) is a two-parameter family of solutions of \(x^{\prime \prime}+\omega^{2} x=0\) on the interval \((-\infty, \infty)\), show that a solution satisfying the initial conditions \(x(0)=x_{0}, x^{\prime}(0)=x_{1}\) is given by $$ x(t)=x_{0} \cos \omega t+\frac{x_{1}}{\omega} \sin \omega t . $$

Verify that the given differential operator annihilates the indicated functions. $$ (D-2)(D+5) ; \quad y=e^{2 x}+3 e^{-3 x} $$

Find the general solution of each differential equation. $$ y^{n}-2 y^{\prime}-2 y=0 $$

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

Verify that the given differential operator annihilates the indicated functions. $$ D^{2}+64 ; \quad y=2 \cos 8 x-5 \sin 8 x $$

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

In Problems 1-36 find the general solution of the given differential equation. \(8 y^{\prime \prime}+2 y^{\prime}-y=0\)

Solve, if possible, the given system of differential equations by either systematic elimination or determinants.\(\begin{aligned} \frac{d x}{d t}+\frac{d y}{d t} &=e^{\prime} \\\\-\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}+x+y &=0 \end{aligned}\)