# Thread: Solving system of nonlinear ODEs

1. ## Solving system of nonlinear ODEs

Hello everyone,

I have a system on the form

$\displaystyle x''_1 + \lambda x'_1 + C(x_1)x_1 = A(x_1)x_3 + B(x_1)x_3^2$

$\displaystyle x'_3 + D(x_1)x_3 + E(x_1)x'_1x_3 = F(x_1)x'_1$

where $\displaystyle A,B,C,D,E,F$ are all nonlinear functions, and both $\displaystyle x_1, x_3$ depend on time.

Does anyone know a good book (if any) which covers methods for the solution of this type of system of nonlinear ODEs? Or will I have to use numerical integration?

Thanks

Update:
Indeed, I need an analytical solution to solve my original problem...

2. Ok... First of all, let $\displaystyle x_3=x_2$ and call the original equations (1), (2) respectively.

Now, (1) can be written as $\displaystyle x_1''+\lambda x_1'=W(x_1,x_2)$
where W is nonlinear in its arguments. By integrating, we obtain $\displaystyle x_1'=F_1(x_1,x_2)$, for a function $\displaystyle F_1$. Substitute this in (2), to end up with $\displaystyle x_2'=F_2(x_1,x_2)$, for a nonlinear function $\displaystyle F_2$.

So, the original system (1)+(2) is only an autonomous system of nonlinear first order ODEs. There is a rich theory behind autonomous systems, and you can get information about the qualitative behaviour of solutions in any good book on ODEs. Now, for obtaining solutions near $\displaystyle x_1=0,x_2=0$ (as I bet is the case) you can turn to a good book on asymptotic methods.

3. Originally Posted by Rebesques
Ok... First of all, let $\displaystyle x_3=x_2$ and call the original equations (1), (2) respectively.

Now, (1) can be written as $\displaystyle x_1''+\lambda x_1'=W(x_1,x_2)$
where W is nonlinear in its arguments. By integrating, we obtain $\displaystyle x_1'=F_1(x_1,x_2)$, for a function $\displaystyle F_1$. Substitute this in (2), to end up with $\displaystyle x_2'=F_2(x_1,x_2)$, for a nonlinear function $\displaystyle F_2$.

So, the original system (1)+(2) is only an autonomous system of nonlinear first order ODEs. There is a rich theory behind autonomous systems, and you can get information about the qualitative behaviour of solutions in any good book on ODEs. Now, for obtaining solutions near $\displaystyle x_1=0,x_2=0$ (as I bet is the case) you can turn to a good book on asymptotic methods.
Is the fact that you've turned the second-order system of ODE's into a system of integro-differential equations important? I mean, "integrating the first equation" sounds nice, but the fact is that you don't yet know what either function is, so your $\displaystyle F_{1}$ function is going to have integrals in it that can't be resolved yet.

4. integro-differential equations

Not really. The functions involved actually define Nemytskii maps. It is a standard trick that I didn't find important to point out.

5. Hmm. Well, you're well beyond my understanding, I'm afraid. Maybe Danny knows something about that.

6. Nemytskii maps - beyond me too! If you let $\displaystyle x_1' = x_2$ then you have a system of 3 first order ODEs. However, the OP want an analytical solution and without knowing the arbitrary functions $\displaystyle A-F$, I don't think we can do that.