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.

$\mathit{y}\mathit{\left(}\mathit{4}\mathit{\right)}\mathit{+}\mathit{8}\mathit{y}\mathit{\text{'}}\mathit{=}{\mathit{4}}^{}$

Short answer

$\mathrm{D}(\mathrm{D}+2)({\mathrm{D}}^2-2\mathrm{D}+4)\mathrm{y}=4^{}$

Step-by-step solution

Use the Differential linear operators to find the solution

We have the non-homogeneous second order differential equation,

$\mathrm{y}\left(4\right)+8\mathrm{y}\text{'}=4^{}$

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

$$\begin{gathered} {\mathrm{D}}^4\mathrm{y}+8\mathrm{Dy}=4 \\ ({\mathrm{D}}^4+8\mathrm{D})\mathrm{y}=4 \end{gathered}$$

After that we have to factorize it as,

$\mathrm{D}({\mathrm{D}}^3+8)\mathrm{y}=4$

After that to expand $({\mathrm{D}}^3+8)$, we have make a long division on (D+2) because (-2) is the number that make $({\mathrm{D}}^3+8)$equals 0, then we have,

$\mathrm{D}(\mathrm{D}+2)({\mathrm{D}}^2-2\mathrm{D}+4)\mathrm{y}=4$

Where g(x)=4

Hence the solution is localid="1667899635906" $\mathbf{D}\mathbf{(}\mathbf{D}\mathbf{+}\mathbf{2}\mathbf{)}\mathbf{(}{\mathbf{D}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathbf{D}\mathbf{+}\mathbf{4}\mathbf{)}\mathbf{y}\mathbf{=}\mathbf{4}$