91Ó°ÊÓ

In Problems, solve the given differential equation by using the substitution \(u=y^{\prime}\). $$ (y+1) y^{\prime \prime}=\left(y^{\prime}\right)^{2} $$

Solve each differential equation by variation of parameters. $$ y^{\prime \prime}+y=\sec ^{2} x $$

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 \(y_{2}(x)\). $$ y^{\prime \prime}-25 y=0 ; \quad y_{1}=e^{5 x} $$

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

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}-8 y^{\prime}+20 y=100 x^{2}-26 x e^{x}\)

In Problems 1-18, solve the given differential equation. $$ x^{2} y^{\prime \prime}+5 x y^{\prime}+3 y=0 $$

Find the general solution of the given second-order differential equation. $$ y^{\prime \prime}-10 y^{\prime}+25 y \quad 0 $$

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} (D+1) x+(D-1) y &=2 \\ 3 x+(D+2) y &=-1 \end{aligned} $$

Solve the given differential equation. $$ x^{2} y^{\prime \prime}+5 x y^{\prime}+3 y=0 $$

In Problems \(3-8\), solve the given differential equation by using the substitution \(u=y^{\prime}\). $$ (y+1) y^{\prime \prime}=\left(y^{\prime}\right)^{2} $$