# How to solve a system of nonlinear 2nd order differential equations?

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Franziska on 21 Feb 2013
Hi there,
I have a challenge solving a system of differential equations 2nd order. I do not receive an error message but have rather strange results... Could you please have a look on what I did and comment on it. If you think the way it was done is fine, I may have the bug some where else...
function dHbar = diffH(tspan,H)
h1 = H(1);
dh1 = H(2);
ddh1 = H(3);
h2 = H(4);
dh2 = H(5);
ddh2 = H(6);
h3 = H(7);
dh3 = H(8);
ddh3 = H(9);
% etc
% the equations are very long. In short the look similar to the following where f(h1,dh1) means some expression as a function of ...
% A1 to A3 are known
ddh1 = ( A1 * f(h1,dh1) + A2 * f(h2,dh2,ddh2) + A3 * f(h3,dh3,ddh3));
ddh2 = ( A2 * f(h2,dh2) + A1 * f(h1,dh1,ddh1) + A3 * f(h3,dh3,ddh3));
ddh3 = ( A3 * f(h3,dh3) + A1 * f(h1,dh1,ddh1) + A2 * f(h2,dh2,ddh2));
dHbar = [h1; dh1; ddh1; h2; dh2; ddh2; h3; dh3; ddh3];
Then I call it with
[T,Hbar] = ode45('diffH',tspan,H); % with tspan - time and H - starting values
As mentioned above, I receive some solution, but it seems strange.
Babak on 4 Mar 2013
I think after your write your equations in the form I'm mentioning above, you can solve it with any ODE solver like ODE45. It doesn't seem to be a complicated case.

Babak on 21 Feb 2013
You need to use the optimization toolbox and it's fsolve(.) routine. You can embed all your nonlinear functions (can be an integrated ODE or whatever)as the arguments of fsolve, assigning thos functions to be the ones that are needed to be optimized.
Franziska on 21 Feb 2013
Thanks for your answer Babak, according to the matlab-help fsolve is not applicable for systems of differential equations. I need the time dependency.
Cheers, Franziska

Franziska on 2 Mar 2013
My code is very sensitive to the choice of initial velocity values. I have some questions here:
1) Is there any possibility to search for appropriate initial velocity values similar to decic for ode15i? 2) Does it make sense if I try with ode15i and decic? Are there any limitations due to singular and time dependent mass matrix? I couldn't find a 2nd order problem similar to my challenge but coded in ode15i which might help me.
Thanks.
Franziska
Alessandro Antonini on 1 Jun 2013
I need to solve a system of 3 equations in the variable x1,x2,x3, I do not know how write the ode function that takes into account a term of a second order derivative of x2 in equation 1. I have a system like that:
ddx1=F1(t)-B1*dx1-M3*ddx3-B3*dx3-M2*ddx2-B2*dx2
ddx2=F2(t)-B2*dx2-K2*dx-M1*ddx1-B1*dx1-M2*ddx2-B2*dx2
ddx2=F3(t)-B3*dx3-K3*dx3-M1*ddx1-B1*dx1-K1*dx1-M2*ddx2-B2*dx2
I do not know how write in the ode function for this system. Can you please explain o write an example of the ode function required to solve a non linear system like that? I would be greateful
Best regards Alessandro Antonini