# Solving second order non-linear equations (time-domain)

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James on 10 Aug 2020
Edited: Alan Stevens on 11 Aug 2020
Hi everyone,
Im starting to learn ODE solving using MATLAB. I'm new to this, although I'm fairly comfortable using MATLAB. (I'm learning alone, as I go)
So, I want to solve an equation like the following one, and basically hoping to have a simulation (sort of) output as a plot. How do I approach this? I'm posting this in advance and I will be looking for resources and other references after posting this :) Again, I'm not sure how to approach this and frankly, I'm not sure how to solve the equation itself.
Apologies for the broad question. Any pointer or lead will be highly appreciated, in a learning point of view.
hosein Javan on 11 Aug 2020
you're welcome.
regarding reference book: "differential equations, boyce and diprima"
you can solve either by homogeneous and particular solutions of second-order ode or by using laplace transform.

Alan Stevens on 10 Aug 2020
Edited: Alan Stevens on 11 Aug 2020
Here's the basic idea. First turn your second-order ODE into twofirst order ones, like so: Then structure your code along the following lines:
P0 = [phi0 v0]; % Initial values of phi and v
tspan = [0 tend]; % Time over which to integrate
[t, P] = ode45(@phidot, tspan, P0); % Call ode solver
phi = P(:,1); % Extract values of phi
plot(t,phi) % plot results
% Function
function dphidt = phidot(t,P)
% Enter data needed e.g. omega4 = etc.
phi = P(1);
v = P(2);
dphidt = [v;
-2nv - wn4^2*phi + etc.];
end
Obviously the above coding is sketchy, but hopefully it's enough to give you the general idea.
James on 11 Aug 2020
This makes a lot of sense. I hope I will be able to figure out the complete code soon, after referring to other notes and sources. Thank you so much for taking your time for this :) Means alot!