The special solution of the differential equation y '' - y = e ^ x satisfying the condition y (0) = 0, y '(0) = 0 is

The special solution of the differential equation y '' - y = e ^ x satisfying the condition y (0) = 0, y '(0) = 0 is

The homogeneous characteristic equation R ^ 2-1 = 0r = ± 1, so the general solution is y = C1E ^ x + c2e ^ (- x). Since the right side of the equal sign is contained in the general solution, let the special solution be y = axe ^ XY '= a (1 + x) e ^ XY' '= a (2 + x) e ^ X. substitute the original equation to get a (2 + x) e ^ x-axe ^ x = e ^ x, and the solution is a = 1 / 2, so the non-homogeneous special solution is y = 1 / 2xe ^ x, so the non-homogeneous general solution is