Solve for the solution of the differential equation and use the method of variation of parameters.
x`` - x = (e^t) + t
v1 = integral of ( g(t) (y1) ) / W
v2 = integral of ( g(t) (y2) ) / W
The Attempt at a Solution
yc= c1 e^t + c2 e^-t
yp = v1 e^t + v2 e^-t
v1 = integral of ((e^-t)(e^t + t)) / -2 = (1/2) (t - te^-t - e^-t)
v2 = integral of ((e^2t)(e^t + t)) / -2 = (1/2) (1/2 e^2t + te^t - e^t)
v1y1 = 1/2 te^t - 1/2 t - 1/2
v2y2 = 1/4 e^t - 1/2t + 1/2
adding these, yp = -t + 1/2 te^t + 1/4 e^t
yc + yp = (c1 e^t + c2 e^-t ) + (-t + 1/2 te^t + 1/4 e^t)
but the answer is (c1 e^t + c2 e^-t ) + (-t + 1/2 te^t)
I got an extra term. Where did I go wrong?