Question

In Problems 1–10 write the given differential equation in the form $\mathbf{L}\mathbf{(}\mathit{y}\mathbf{)}\mathbf{=}\mathit{g}\mathbf{(}\mathit{x}\mathbf{)}$where L is a linear differential operator with constant coefficients. If possible, factor L.$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathit{y}\mathbf{\text{'}}\mathbf{-}\mathbf{12}\mathit{y}\mathbf{=}\mathit{x}\mathbf{-}\mathbf{6}$

Short answer

$(D+2)(D-6)y=x-6$

Step-by-step solution

Use the Differential linear operators to find the solution  

We have the non-homogeneous second order differential equation,

$y\text{'}\text{'}-4y\text{'}-12y=x-6$

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

$$\begin{gathered} D^{2}y-4Dy-12y=x-6 \\ (D^2-4D-12)y=x-6 \end{gathered}$$

After that we have to factorize it as,

$(D+2)(D-6)y=x-6$

Where.$g\left(x\right)=x-6$

Hence the solution isrole="math" localid="1664208196519" $\mathbf{(}\mathit{D}\mathbf{+}\mathbf{2}\mathbf{)}\mathbf{(}\mathit{D}\mathbf{-}\mathbf{6}\mathbf{)}\mathit{y}\mathbf{=}\mathit{x}\mathbf{-}\mathbf{6}$