91Ó°ÊÓ

Solve the given differential equation by undetermined coefficients. \(\frac{1}{4} y^{\prime \prime}+y^{\prime}+y=x^{2}-2 x\)

In Problems \(1-16\), the indicated function \(y_{1}(x)\) is a solution of the given equation. Use reduction of order or formula (5), as instructed, to find a second solution \(y_{2}(x)\). $$ y^{\prime \prime}-y=0 ; \quad y_{1}=\cosh x $$

In Problems 1-18, solve the given differential equation. $$ x^{2} y^{\prime \prime}+x y^{\prime}+4 y=0 $$

In Problems, solve the given differential equation by using the substitution \(u=y^{\prime}\). $$ x^{2} y^{\prime \prime}+\left(y^{\prime}\right)^{2}=0 $$

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

Given that \(y=c_{1}+c_{2} x^{2}\) is a two-parameter family of solutions of \(x y^{\prime \prime}-y^{\prime}=0\) on the interval \((-\infty, \infty)\), show that constants \(c_{1}\) and \(c_{2}\) cannot be found so that a member of the family satisfies the initial conditions \(y(0)=0, y^{\prime}(0)=1\). Explain why this does not violate Theorem 3.1.1.

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\left(D^{2}+5\right) x-2 y=0 \\ &-2 x+\left(D^{2}+2\right) y=0 \end{aligned} $$

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\left(D^{2}+5\right) x-2 y=0 \\ &-2 x+\left(D^{2}+2\right) y=0 \end{aligned} $$

In Problems \(3-8\), solve the given differential equation by using the substitution \(u=y^{\prime}\). $$ x^{2} y^{\prime \prime}+\left(y^{\prime}\right)^{2}=0 $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}-8 y^{\prime}+20 y=100 x^{2}-26 x e^{x} $$