Answer to Question #212967 in Differential Equations for mike

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

Yield is given by, $\frac{dy}{dt}+y=100+e^{-t}$

$I.F. e^{\int dt} =e^t$

Multiplying by Integrating Factor both sides, and integrating

$e^{t}(\frac{dy}{dt}+y)=e^{t}(100+e^{-t})$

$ye^{t} = \int (100 e^{t} + 1) dt$

$ye^{t} = 100 e^{t} + t + C$

Applying condition y(0) = 0,

$0e^{0} = 100 e^{0} + 0 + C \implies C = -100$

$ye^{t} = 100 e^{t} + t -100$

So, yield as a function of t is given by

$y = 100 + t e^{-t}-100e^{-t}$

$y = 100 (1-e^{-t})+ t e^{-t}$