The example: (1) x''(t) + y(t) = cos(t) (2) x(t) = cos(t) is trivially solvable by differentiating (2) twice and substituting into (1) to give: -

The example: (1) x''(t) + y(t) = cos(t) (2) x(t) = cos(t) is trivially solvable by differentiating (2) twice and substituting into (1) to give: -cos(t) + y(t) = cos(t) or y(t) = 2 cos(t) Together with (2) that gives the solution; there is no need to use advanced techniques to solve this particular example.