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Given $K(0) = 0,2P$. I'm supposed to solve the ODE

$ \frac{dK}{dt} = \lambda K(P-K)$

I have tried to seperate and integrate both sides

$ \int \frac{1}{K(P-K)} dK = \int \lambda \space dt$

to get

$ \ln|K(P-K)| = \lambda t + C$

and then solve for $K$

$ e^{\ln|K(P-K)|} = K(P-K)=e^{\lambda t + C}$

But there I'm stuck as to getting any further to finding the general solution. Does the $K(P-K)$ term require integrating using partial fractions?

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    At the point at which you say you are stuck, you have a quadratic equation for $K$. Surely you can solve a quadratic equation? You'll still need to do this once you've corrected the integration.2012-10-31

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Yes, you need partial fractions. See this question about partial fractions. Write $\frac{1}{{K(P - K)}} = \frac{a}{K} + \frac{b}{{(P - K)}}$ and solve for $a$ and $b$. If the link does not help, Google it. It is not as difficult as it might seem, especially in this case.