the original system is $$\begin{cases}\dot{x} = y +\cos t \\ \dot{y} = 1 -x\end{cases}$$

it does not look like homogeneous one, so I do not know how to proceed with it. I have never worked wit it.

as for homogeneous I know that we have to:

1.) write down the matrix from the given system

2.) find its eigenvalues and therefore eigenvectors/ generalized eigenvectors

3.) write diwn the answer

is there big difference between homogeneous and non-homogeneous systems in solving proccess? And I am also confused about $\cos t$ a lot

  • 1
    $\begingroup$ $\ddot{y}=-\dot{x}=-y-\cos t$, so $\ddot{y}+y=-\cos t$. Can you solve this? $\endgroup$
    – A. Goodier
    Dec 23, 2017 at 17:31
  • $\begingroup$ Are you familiar with Laplace transform? $\endgroup$
    – Nosrati
    Dec 23, 2017 at 17:36
  • $\begingroup$ @MyGlasses, no, our professor did not explain it to us yet $\endgroup$
    – M.Mass
    Dec 23, 2017 at 17:43
  • $\begingroup$ @woofy, which method did you use to get this? $\endgroup$
    – M.Mass
    Dec 23, 2017 at 18:07
  • 1
    $\begingroup$ @M.Mass yes, see chapter 26 p.270 of James Robinson's book on ODEs here: faculty.mu.edu.sa/public/uploads/… $\endgroup$
    – A. Goodier
    Dec 23, 2017 at 20:29

1 Answer 1


write $$x(t)=1-y'(t)$$ so we have $$x'(t)=-y''(t)$$ plugging this in your first equation you have to solve $$-y''(t)=y(t)+\cos(t)$$


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