Question

1.sketch the gradient field for the differential equation dy/dt=t-y² for t=-1......,1

y=-1,.......,1

2.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

Part 1

Part 2

$y'+f(x)y=g(x)$ as $I(x) = e^{\int f(x)dx}\\$

Given that the yield y(t) of a corn crop satisfies the equation $\frac{dy}{dt}=100+e^{-t}-y$

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

Calculating the integrating factor

$I(t) = e^{\int f(t)dt} = e^t$

Multiplying both sides

$e^t\frac{dy}{dt}+ye^t=100e^{-t}+1\\ (e^ty)'=100e^t+1\\ \int(e^ty)'dt=\int(100e^t+1)\\ e^ty=\int100e^tdt+\int1dt\\ e^ty=100e^t+t+C\\ y= 100+te^{-t}+Ce^{-t}\\ 0= 100+0*e^{0}+Ce^{0}\\ C=-100\\ y= 100+te^{-t}-100e^{-t}\\$