Question
Solve this differential equation.
dy/dt=0.07y(1-y/100), y(0)=10
dy/dt=0.07y(1-y/100), y(0)=10
Answers
Answered by
oobleck
dy/dt=0.07y(1-y/100)
dy/(y(1-y/100)) = .07 dt
(1/y + 1/(100-y)) dy = .07 dt
ln(y/(100-y)) = .07t + C
y/(100-y) = c e^(.07t)
y = ce^(.07t) - ce^(.07t)y
y(1+ce^(.07t)) = ce^(.07t)
y = ce^(.07t)/(1+ce^(.07t))
y = 1/(1+ce^(-.07t))
since y(0) = 10, we have
1/(1+c) = 10
c = -9/10
y = 1/(1 - 9/10 e^(-.07t))
Hmmm. wolframalpha got
y = 100/(1 + 9e^(-.07t))
Better check my math.
dy/(y(1-y/100)) = .07 dt
(1/y + 1/(100-y)) dy = .07 dt
ln(y/(100-y)) = .07t + C
y/(100-y) = c e^(.07t)
y = ce^(.07t) - ce^(.07t)y
y(1+ce^(.07t)) = ce^(.07t)
y = ce^(.07t)/(1+ce^(.07t))
y = 1/(1+ce^(-.07t))
since y(0) = 10, we have
1/(1+c) = 10
c = -9/10
y = 1/(1 - 9/10 e^(-.07t))
Hmmm. wolframalpha got
y = 100/(1 + 9e^(-.07t))
Better check my math.
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