Question: Identify each of the differential equations in Problems 1to 24 as to type (for example, separable, linear first order, linear second order, etc.), and then solve it.

${\mathbf{(}\mathbf{D}\mathbf{-}\mathbf{2}\mathbf{)}}^{\mathbf{2}}\left(D^2+9\right)\mathit{y}\mathbf{=}\mathbf{0}$

The differential equation is ${(D-2)}^2\left(D^2+9\right)y=0$.

When fand its derivatives are inserted into the equation, a solution is a function y = f(x) that solves the differential equation. The highest order of any derivative of the unknown function appearing in the equation is the order of a differential equation.

A differential equation of the form$\mathbf{(}\mathit{D}\mathbf{-}\mathit{a}\mathbf{)}\mathbf{(}\mathit{D}\mathbf{-}\mathit{b}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{a}\mathbf{\ne }\mathit{b}$ has general solution$\mathit{y}\mathbf{=}{\mathit{c}}_{\mathbf{1}}{\mathit{e}}^{\mathbf{a}\mathbf{x}}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathit{e}}^{\mathbf{b}\mathbf{x}}\mathbf{.}$

Consider the equation.

${(D-2)}^2\left(D^2+9\right)y=0$

The auxiliary equation is,

$\begin{array}{l}{(m-2)}^2\left(m2+9\right)=0\\ m=2,2,-3i,3i\end{array}$

The solution for m= 2, 2 is,

$y_1=\left(c_{1}x+c_2\right)e^{2x}$

The solution for$m=-3i,3i$ is,

$\begin{array}{l}{y}_2=e^0\left[C_3\cos x+C_4\sin x\right]\\ =C_3\cos x+C_4\sin x\end{array}$

The solution of differential equation is,

$\begin{array}{l}y=y_1+y_2\\ y=\left(c_{1}x+c_2\right)e^{2x}+C_3\cos x+C_4\sin x\end{array}$

Thus, the solution of given differential equation is ,

$y=\left(c_{1}x+c_2\right)e^{2x}+C_3\cos x+C_4\sin x.$