Question

In Problems 1–10 write the given differential equation in the form where L is a linear differential operator with constant coefficients.

If possible, factor L.

$\mathit{y}\mathbf{"}\mathbf{\text{'}}\mathbf{+}\mathbf{10}\mathit{y}\mathbf{"}\mathbf{+}\mathbf{25}\mathit{y}\mathbf{\text{'}}\mathbf{=}{\mathit{e}}^{\mathbf{x}}$

Short answer

$D{(D+5)}^{2}y=e^x.$

Step-by-step solution

Use the Differential linear operators to find the solution

We have the non-homogeneous second order differential equation,

$y"\text{'}+10y"+25y\text{'}=e^x$

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

$$\begin{gathered} D^{3}y+10D^{2}y+25Dy=e^x \\ (D^3+10D^2+25D)y=e^x \end{gathered}$$

After that we have to factorize it as,

$$\begin{gathered} D(D^2+10D+25)y=e^x \\ D(D+5)(D+5)y=e^x \\ D{(D+5)}^{2}y=e^x \end{gathered}$$

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

Hence the solution is$\mathit{D}{\mathbf{(}\mathit{D}\mathbf{+}\mathbf{5}\mathbf{)}}^{\mathbf{2}}\mathit{y}\mathbf{=}{\mathit{e}}^x$