91Ó°ÊÓ

Each of Problems 43 through 48 gives a general solution \(y(x)\) of a homogeneous second-order differential equation \(a y^{\prime \prime}+b y^{\prime}+c y=0\) with constant coefficients. Find such an equation. $$ y(x)=c_{1}+c_{2} x $$

Each of Problems 43 through 48 gives a general solution \(y(x)\) of a homogeneous second-order differential equation \(a y^{\prime \prime}+b y^{\prime}+c y=0\) with constant coefficients. Find such an equation. $$ y(x)=e^{x}\left(c_{1} e^{x \sqrt{2}}+c_{2} e^{-x \sqrt{2}}\right) $$

Solve the initial value problem $$ y^{(3)}=y ; \quad y(0)=1, \quad y^{\prime}(0)=y^{\prime \prime}(0)=0 . $$ (Suggestion: Impose the given initial conditions on the general solution $$ y(x)=A e^{x}+B e^{\alpha x}+C e^{\beta x} $$ where \(\alpha\) and \(\beta\) are the complex conjugate roots of \(r^{3}-1=\) 0 , to discover that $$ y(x)=\frac{1}{3}\left(e^{x}+2 e^{-x / 2} \cos \frac{x \sqrt{3}}{2}\right) $$ is a solution.)

Solve the initial value problem $$ \begin{aligned} &y^{(4)}=y^{(3)}+y^{\prime \prime}+y^{\prime}+2 y \\ &y(0)=y^{\prime}(0)=y^{\prime \prime}(0)=0,2 y^{(3)}(0)=30 \end{aligned} $$

Use the method of variation of parameters to find a particular solution of the given differential equation. $$ y^{\prime \prime}-4 y^{\prime}+4 y=2 e^{2 x} $$

Use the method of variation of parameters to find a particular solution of the given differential equation. $$ y^{\prime \prime}-4 y=\sinh 2 x $$

Make the substitution \(v=\ln x\) of Problem 51 to find general solutions (for \(x>0\) ) of the Euler equations in Problems. $$ x^{2} y^{\prime \prime}+x y^{\prime}+9 y=0 $$

Use the method of variation of parameters to find a particular solution of the given differential equation. $$ y^{\prime \prime}+9 y=2 \sec 3 x $$

Use the method of variation of parameters to find a particular solution of the given differential equation. $$ y^{\prime \prime}+4 y=\sin ^{2} x $$

Use the method of variation of parameters to find a particular solution of the given differential equation. $$ y^{\prime \prime}-4 y=x e^{x} $$