# Autonomous nonlinear ODE

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?

• Yes, you need partial fractions. What you wrote after your integral is incorrect. In particular, $$\frac{1}{P} \left[ \frac{1}{K} + \frac{1}{P-K} \right] = \frac{1}{K(P-K)}$$. Oct 31 '12 at 0:04
• 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. Oct 31 '12 at 0:14

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.