Answer to Question #213427 in Calculus for sam

Question

The yield y(t) (in bushes)per acre of a corn crop satisfies the equation dy/dt+y=100+e^-t

if y(0)=0 find y at any time t

Answer 1

$y'+y=100+e^{-t}$

Integrating factor: $\mu=e^t$

$(e^ty)'=100e^t+1$

$e^ty=\int(100e^t+1)dt$

$e^ty=100e^t+t+c$

$y(t)=100+te^{-t}+ce^{-t}$

$y(0)=0$

$0=100+0+ce^{-0}=>c=-100$

$y(t)=100+te^{-t}-100e^{-t}$