- #1
Luminous Blob
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I'm doing some revision on differential equations, and am getting stuck on what should be a simple problem.
The question is:
Solve: dP/dt = 0.2P(1000 - P)
Find particular solutions when (i) P(0) = 1000 and (ii) P(0) = 2000
The answers are supposed to be:
i) P(t) = 1000
ii) P(t) = 1000[1 - (1/2)exp(-200t)]^(-1)
I keep getting the wrong answers. The general solution I came up with (after doing the partial fraction decomposition and integrating) is:
t + C = 1/200[ ln|P| - ln|1000 - P| ]
exp(200t + C) = exp(ln|P| - ln|1000 - P|)
Aexp(200t) = P/(1000 - P) {where A = exp(C)}
P = 1000Aexp(200t) - (P)Aexp(200t)
P[ 1 + Aexp(200t) ] = 1000Aexp(200t)
P = 1000Aexp(200t)/[ 1 + Aexp(200t)]
But I don't get the right answers when I solve for A with the initial condition.
I'd appreciate it if anyone can tell me where I'm going wrong here.
The question is:
Solve: dP/dt = 0.2P(1000 - P)
Find particular solutions when (i) P(0) = 1000 and (ii) P(0) = 2000
The answers are supposed to be:
i) P(t) = 1000
ii) P(t) = 1000[1 - (1/2)exp(-200t)]^(-1)
I keep getting the wrong answers. The general solution I came up with (after doing the partial fraction decomposition and integrating) is:
t + C = 1/200[ ln|P| - ln|1000 - P| ]
exp(200t + C) = exp(ln|P| - ln|1000 - P|)
Aexp(200t) = P/(1000 - P) {where A = exp(C)}
P = 1000Aexp(200t) - (P)Aexp(200t)
P[ 1 + Aexp(200t) ] = 1000Aexp(200t)
P = 1000Aexp(200t)/[ 1 + Aexp(200t)]
But I don't get the right answers when I solve for A with the initial condition.
I'd appreciate it if anyone can tell me where I'm going wrong here.