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: $\mu=e^t$

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

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

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

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

$y(0)=0$

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

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