Solve the given differential equation by separation of variables.
dP/dt=p-p^2
Ans:
dP / (P - P^2) = dt
dP/(P(1 - P)) = dt
dP/(P(P - 1)) =-dt
A/P + B/(P-1) = 1/(P(P-1))
A(P-1) + BP = 1
(A+B)P - A = 1
So, A = -1
Thus, B = 1
So, it becomes :
-1/(P) + 1/(P-1) * dP = -dt
ln|P - 1| - ln|P| = -t + C
ln|(P - 1)/P| = -t + C
ln|1 - 1/P| = -t + C
1 - 1/P = Ce^(-t)
1/P = 1 - Ce^(-t)
P = 1 / (1 - Ce^(-t))
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