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.

${\mathbf{y}}^{\mathbf{\left(}\mathbf{4}\mathbf{\right)}}\mathbf{-}\mathbf{8}\mathbf{y}\mathbf{"}\mathbf{+}\mathbf{16}\mathbf{y}\mathbf{=}\mathbf{(}{\mathbf{x}}^{\mathbf{3}}\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{)}{\mathbf{e}}^{\mathbf{4}\mathbf{x}}$

Short answer

${(\mathrm{D}-2)}^2{(\mathrm{D}+2)}^2\mathrm{y}=({\mathrm{x}}^3-2\mathrm{x}){\mathrm{e}}^{4\mathrm{x}}$

Step-by-step solution

Use the Differential linear operators to find the solution

We have the non-homogeneous second order differential equation,

${\mathrm{y}}^{\left(4\right)}-8\mathrm{y}"+16\mathrm{y}=({\mathrm{x}}^3-2\mathrm{x}){\mathrm{e}}^{4\mathrm{x}}$

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

$$\begin{gathered} {\mathrm{D}}^4\mathrm{y}-8{\mathrm{D}}^2\mathrm{y}+16\mathrm{y}=({\mathrm{x}}^3-2\mathrm{x}){\mathrm{e}}^{4\mathrm{x}} \\ ({\mathrm{D}}^4-8{\mathrm{D}}^2+16)\mathrm{y}=({\mathrm{x}}^3-2\mathrm{x}){\mathrm{e}}^{4\mathrm{x}} \end{gathered}$$

After that we have to factorize it as,

localid="1667899323559" $$\begin{gathered} ({\mathrm{D}}^2-4)({\mathrm{D}}^2+4)\mathrm{y}=({\mathrm{x}}^3-2\mathrm{x}){\mathrm{e}}^{4\mathrm{x}} \\ (\mathrm{D}-2)(\mathrm{D}+2)(\mathrm{D}-2)(\mathrm{D}+2)\mathrm{y}=({\mathrm{x}}^3-2\mathrm{x}){\mathrm{e}}^{4\mathrm{x}} \\ {(\mathrm{D}-2)}^2{(\mathrm{D}+2)}^{2}y=({\mathrm{x}}^3-2\mathrm{x}){\mathrm{e}}^{4\mathrm{x}} \end{gathered}$$

Where$\mathrm{g}\left(\mathrm{x}\right)=({\mathrm{x}}^3-2\mathrm{x}){\mathrm{e}}^{4\mathrm{x}}$

Hence the solution is localid="1667899373915" ${\mathbf{(}\mathbf{D}\mathbf{-}\mathbf{2}\mathbf{)}}^{\mathbf{2}}{\mathbf{(}\mathbf{D}\mathbf{+}\mathbf{2}\mathbf{)}}^{\mathbf{2}}\mathbf{y}\mathbf{=}\mathbf{(}{\mathbf{x}}^{\mathbf{3}}\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{)}{\mathbf{e}}^{\mathbf{4}\mathbf{x}}$