Homework Sourseslope fields problem for the equation dPdt00333333P30P for P
(slope fields problem) for the equation dP/dt=0.0333333P(30-P), for P bigger than or equal to 0
What is the equation of the solution to this differential equation that passes through (0,0)?
P=
please show work.
What is the equation of the solution to this differential equation that passes through (0,0)?
P=
please show work.
Solution
Given equation is dP/dt = 1/30 [P(30-P)]
[1/P + 1/(30-P)]dP/dt = 1
Hence dp [ 1/P + 1/(30-P)] = dt
Integration both sides
[Log(p(t))-Log(30-p(t))] = [t] + k
Log[p(t)/(30-p(t)] = t + Log(k) ...... Log(k) is constant of integration
p/(30-p) = k*e^t
=> p(t) = 30*k*e^t/(1+ke^t) is the general solution.
at t=0, p=0 => the solution of the equation passing thru (0,0) is given by k=0 ie
p(t) = 0 for all t
