Solve this system of odes by eliminating y.

let d = y' , d² = y'', d³ = y''' etc.

d²x - dy = - 6t

dx + dy = y - x + 3

Adding the two ODEs gives



Differentiating once gives




The first ODE lets you simplify this a bit:




Let , so that , and reduce the order of the ODE to get



Solving this ODE for shouldn't be a problem for you; you would find that



Since , integrating once with respect to gives the general solution for :



Now plug in the second derivative of this solution into the first ODE to find another ODE in alone.






Integrate with respect to and you get