# Solution to first-order non-linear system of ordinary differential equations

1. Can someone help me solve the following system of differential equations?

$$\begin{eqnarray*} \frac{dP_0}{dt}& = & -C_1\lambda P_0 P_1+ \frac{C_1}{2}P_1^2 \\ \frac{dP_1}{dt} &= &-C_2P_1+ C_1 \lambda P_0 P_1 - \frac{1}{2} (1+ \lambda)C_1P_1^2+ C_1P_1P_2 \\ \frac{dP_2}{dt}& = &C_2P_1+ \frac{C_1}{2} \lambda P_1^2 - C_1P_1P_2 \end{eqnarray*}$$

1. (Extending above) Suppose we have $$K$$ first order non-linear differential equations (similar to above), is there any simple method to solve them analytically?

1. If you will summarize all three equations, you get $$\frac{dP_0}{dt}+\frac{dP_1}{dt}+\frac{dP_2}{dt} = 0,$$ and therefore, $$\frac{d(P_0 + P_1 + P_2)}{dt} = 0,$$ and therefore, $$P_0 + P_1 + P_2 = C,$$ where $C$ is some constant.
Try now to substitute, say, $P_0 = C - P_1 - P_2$ to your system. Simplifying the expressions you will get the (algebraic, but not differential) dependence between $P_1$ and $P_2$. Substituting this dependence again, you will obtain one ODE of the first order, which will be easy to solve.