# How to solve a system of coupled, non-homogenous differential equations of motions?

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Jake il 17 Nov 2022
Commentato: Torsten il 18 Nov 2022
Hi,
First, I must say that I'm still learning the mathematics behind the governing equations, and I know I have a long way to go. But I cannot even begin to understand how one can go about solving the system of equations of motion. I added the following to express the type of equation system that I'm trying to solve. Again, it is not the complete system, and I want to learn how to approach it.
Suppose (a_1 - a_7) and b_1, b_2 are known. The (0) here means that it is the value at "zero frequency" . Further, the coefficients are known for a series of frequencies. So, I think the system of equations are to be solved at each frequency using the known coefficients. I have not included the complete system, and the last equation is also not complete. I was simply explaining the structure. But I can add the complete system here as well.
I looked through a lot of questions that were asked here and followed some tutorials too. But I still cannot understand how to go about this. I would really appreciate any help or advice (or any links to read/learn more). My apologies if the question is irrelevant here, but I’m really helpless at this point.
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### Risposte (1)

Torsten il 17 Nov 2022
Modificato: Torsten il 17 Nov 2022
Do you know how to transform a system of 2nd order equations to a system of first order equations ?
So this is the first step:
Set
x = x1, xdot = x1p, y = y1, ydot = y1p, phi = phi1, phidot = phi1p, psi = psi1, psidot = psi1p,...
dx1/dt - x1p = 0
dy1/dt - y1p = 0
dphi1/dt - phi1p = 0
dpsi1/dt - psi1p = 0
a1(0)*dx1p/dt + a2(0)*y1p*psi1p + a3(0)*phi1p*psi1p + a4(0)*psi1p^2 - F1 = 0
...
(same for equations 2 and 3).
Now you have a 1st-order system of differential equations of the form
F(z,zdot,t) = 0
with vectors z and zdot as
z = [x1 x1p y1 y1p phi1 phi1p psi1 psi1p ...]
zdot = [dx1/dt dx1p/dt dy1/dt dy1p/dt dphi1/dt dphi1p/dt dpsi1/dt dpsi1p/dt ...]
Use ODE15I to solve it.
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Jake il 18 Nov 2022
Only known terms are the coefficients and the initial conditions. i.e. Initial conditions of the six velocities (3 linear, and 3 rotational) are known.
(Say, for instance, xdot0=1 and all other velocities ydot0, zdot0, phidot0, thetadot0, psidot0 = 0). In my head, I thought this would be an iterative process but I have no idea anymore :(
Torsten il 18 Nov 2022
I have no idea anymore :(
Me too. Force terms (F1, F2,...) are usually given and not unknowns.

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