We are asked to find f from the data f ''(t) = 5et + 3 sin t where f(0) = 0, f(π) = 0. We determine first f'(t) using integral calculus. f'(t) = 5 et - 3 cos t + c
f(t) = 5 et -3 sin t + ct + d
f(0) = 0 = 5 + d
d = -5
f(π) = 0 = 5 e pi + c pi + d
c is equal to -(5 e pi+d) / pi
The answer is f(t) = 5 et -3 sin t -(5 e pi+d) t/ pi - 5
