# solving coupled system of second order differential equations

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aakash dewangan on 25 May 2021
Commented: aakash dewangan on 30 May 2021
Hello everyone,
I want to solve a "second order coupled ordinary differential equation". I searched a lot but could not find the solution.
Please suggest me how can I solve this.
The structure of my equation is given below,
[M]{x''} + [K]{x} = {F}
where [M], [K] are the matrices, which contain time dependent terms.
{x} vector of unknown dependent variables.
{x''} is the second derivative of the vector {x} with respect to time.
Please note that [M], [K] contains time varying terms
Looking forward for your the response.
Paul on 28 May 2021
Do you have a simple example for M, K, and F? Preferably one that you know what that solution should be?
aakash dewangan on 30 May 2021
M, K and F contains sinusoidal terms, which depend on time. Diagonal terms of M and K are in the form of a+sin(nwt), and non diagonal terms are like sin(nbt)*sin(nct). Where a,b,c,n are some constant parameters, and t is time.

Sulaymon Eshkabilov on 25 May 2021
Hi,
You can employ ode solvers (ode23, ode23tb, ode45, ode113, etc) as suggested or write scripts using function handles or anonymous functions by apply Euler or Runge-Kutta methods.
aakash dewangan on 28 May 2021
Thanks Sulaymon,
But I am looking for Analytical solution. Can you suggest me how I can solve this using Analytical approach?
Sulaymon Eshkabilov on 28 May 2021
Should you need to obtain an analytical solution, then dsolve() of Symbolic MATH toolbox needs to be employed. E..g.:
syms x(t) Dx(t) DDx(t)
Dx = diff(x, t);
DDx = diff(Dx, t);
M = [??];
K = [??];
EQN = DDx==inv(M)*(F-K*x);
SOL = dsolve(EQN, x(0)==??, Dx(0)==??)
Good luck