1
$\begingroup$
  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?
$\endgroup$
1
$\begingroup$
  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.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.