# 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 Answer

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.

1. According to your second question. The answer is negative. There is no any general method to deal with non-linear problems, since the structure of them can be very diverse and complex.