91Ó°ÊÓ

In Problems 1-24 find a second solution of each differential equation. Use reduction of order or formula (5) as instructed. Assume an appropriate interval of validity. $$ y^{\prime \prime}-y=0 ; \quad y_{1}=\cosh x $$

Solve the given differential equation by undetermined coefficients. $$ y^{*}+3 y=-48 x^{2} e^{3 x} $$

In Problems 1-24 find a second solution of each differential equation. Use reduction of order or formula (5) as instructed. Assume an appropriate interval of validity. $$ y^{\prime \prime}-25 y=0 ; \quad y_{1}=e^{5 x} $$

Write the given differential equation in the form \(L(y)=g(x)\), where \(L\) is a linear differential operator with constant coefficients. If possible, factor \(L\). $$ y^{\prime \prime \prime}+4 y^{\prime \prime}+3 y^{\prime}=x^{2} \cos x-3 x $$

Solve the given differential equation by using the substitution \(u=y^{\prime}\). $$ y^{2} y^{\prime \prime}=y^{\prime} $$

Solve, if possible, the given system of differential equations by either systematic elimination or determinants.\(\frac{d^{2} x}{d t^{2}}+\frac{d y}{d t}=-5 x\) \(\frac{d x}{d t}+\frac{d y}{d t}=-x+4 y\)

Solve each differential equation by variation of parameters. State an interval on which the general solution is defined. $$y^{\prime \prime}-y=\sinh 2 x$$

Solve the given differential equation. \(x^{2} y^{\prime \prime}+3 x y^{\prime}-4 y=0\)

Solve the given differential equation by undetermined coefficients. $$ 4 y^{\prime \prime}-4 y^{\prime}-3 y=\cos 2 x $$

In Problems 1-36 find the general solution of the given differential equation. \(y^{\prime \prime}-3 y^{\prime}+2 y=0\)