Solve the differential equation => dr / dt = [ (1 + e ^ t ) ^ 2 ] / e ^ t...........?

Answer 1

Rearrange and integrate:

r = ∫ (1+e^t)²/e^t dt = 2(t + sinh t) + C

Answer 2

If you call 1+e^t= z e^t dt= dz
dt=dz/(z-1)
Int z^2/(z-1)^2 dz = Int (1+(2z-1)/(z-1)^2 dz = z +2lnIz-1I-1/(z-1)

r= e^t+2t-1/e^t +C
(Please verify calculations)