Question

A calf that weighs x pounds at birth gains weight at the rate of dw/dt = 1200-w, where w is weight in pounds and t in time in years. Solve the differential equation.

Answer 1

w' = 1200 − w
w'⋅(e^t) + w⋅(e^t) = 1200⋅(e^t)
w'⋅(e^t) + w⋅(d/dx e^t) = 1200⋅(e^t)
d/dt [ w⋅(e^t) ] = 1200⋅(e^t)
∫ d/dt [ w⋅(e^t) ] dt = ∫ 1200⋅(e^t) dt
w⋅e^t = 1200⋅e^t + C
w = [ 1200⋅e^t + C ] ⁄ e^t
w = 1200 + C⋅e^(−t)

w_o = w(0) = 1200 + C⋅e^(−0) = 1200 + C
C = w_o − 1200

w = 1200 + (w_o − 1200)⋅e^(−t)