Solving system of differential equations $ D\,x+z=e^t ,(D-1)\,x+D\,y+D\,z=0 , x+2\,y+D\,z=e^t $

Question

Solve the given system of differential equations by systematic elimination. \begin{cases} D\,x+z=e^t \\ (D-1)\,x+D\,y+D\,z=0 \\ x+2\,y+D\,z=e^t \end{cases}

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My process of solving: $$ y= \frac{(1-2D)\,}{(D+2)} \,z $$ $$ x=\frac{(D^2-3D)}{(D+2)(D+1)}\, z $$ However I wasn't able to do rest, the problems seems very interesting! This is my first time doing differential equation in $(x(t), y(t), z(t)) $ system. I appreciate the community to help me.

Answer 1

Extract $z(t)$ from the first equation and $y(t)$ from the third.

Plug all of that in the second equation; you should have an equation of third order in $x(t)$ with constant coefficients. It is simple to solve.