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

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.

Thanks in advance!

[hosein Javan]

if "w_n4^2" and "n" are time-independant then your ode is linear inhomogeneous. in this case the solution is analytically found. you only have to program the general and particular solution. if you don't want to bother solving it, you can use symbolic tollbox. otherwise if if "w_n4^2" and "n" are time-dependant, then you have to use "ode45", "ode23", ...

but I recommend use simulink because it is a powerful and fast ode solver with many settings and optimizations.

if you have any questions regarding these,...

[Jake]

Thank you so much for the response!

Although I'm not familiar with the Symbolic toolbox, I'm fairly good with Simulink. I think I will try that. I understand that I have to study the math as well as solving it using MATLAB :) I was digging through the books too.

> if you have any questions regarding these,

I'm not sure if you were going to add some links here. I'd love to see some references though, if you have ay recommendations :)

Thank you again!

[hosein Javan]

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.

Answer 1

[Alan Stevens]

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.

[Jake]

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!