Question

Find a formula involving integrals for a particular solution of y'''-y''+y'-y=g(t).

Answer 1

Let \(y=e^{kt}\)
$$y'''-y''+y'-y=0\\k^3e^{kt}-k^2e^{kt}+ke^{kt}-e^{kt}=0\\k^3-k^2+k-1=0\\k^2(k-1)+1(k-1)=0\\(k-1)(k^2+1)=0\\k=1,\pm i$$hence \(\{e^t,e^{it},e^{-it}\}\) form a basis for our solution space and we have general solutions

Answer 2

Now, you can use the method of variation of parameters to write a particular solution in terms of these.

Answer 3

@oldrin.bataku that was bomb