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## Homework Statement

Solve for the solution of the differential equation and use the method of variation of parameters.

x`` - x = (e^t) + t

## Homework Equations

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W= (y2`y1)-(y2y1`)

v1 = integral of ( g(t) (y1) ) / W

v2 = integral of ( g(t) (y2) ) / W

## The Attempt at a Solution

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yc= c1 e^t + c2 e^-t

yp = v1 e^t + v2 e^-t

W= -2

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?

Thank you!