Question

In Problems 1–10 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.

$\mathit{y}\mathbf{"}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathit{y}\mathbf{\text{'}}\mathbf{=}{\mathit{e}}^{\mathbf{x}}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathbf{2}\mathit{x}$

Short answer

$D\left(D^2+4\right)y=e^x\cos 2x.$

Step-by-step solution

Use the Differential linear operators to find the solution

We have the non-homogeneous second order differential equation,

$y"\text{'}+4y\text{'}=e^x\cos 2x$

And we have to write it in the form L(y) = g(x) using the differential linear operators

as

$$\begin{gathered} D^{3}y+4Dy=e^x\cos 2x \\ \left(D^3+4D\right)y=e^x\cos 2x \end{gathered}$$

After that we have to factorize it as,

$D\left(D^2+4\right)y=e^x\cos 2x$

Where $g\left(x\right)=e^x\cos 2x$

Hence the solution is $\mathit{D}\left(D^2+4\right)\mathit{y}\mathbf{=}{\mathit{e}}^{\mathbf{x}}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathbf{2}\mathit{x}$