# Matlab: Solving two second order differential equations

## Homework Statement

Write a function and solver for the chaotic motion of stars in a galaxy

Initial conditions:

x at time zero = 0
velocity of x at time zero = 0.3
y at time zero = 0
velocity of y at time zero = 0

Time period 2 to 200 with an interval od 0.02

## Homework Equations

U(x,y)=x^2y-y^3/3

This gives rise to the equations:

d^2x/dt^2= -x-2xy;

d^2y/dt^2= -y-x^2+y^2.

And this gives rise to the four first-order equations

dx/dt=velocity of x

dvelocity of y/dt = -x -2xy

dy/dt = velocity of y

dvelocity of y/dt = -y=x^2+y^2

## The Attempt at a Solution

This is my first time learning matlab and all the examples we were shown until now dealt with first-ODEs. I've made lots and lots of attempts at the solution, all coming up with all kinds of errors like undefined variables on Matlab. Basically my problem is that I can't figure out how to structure the function and solver for so many different terms.

One hopeless attempt was:

Denoting:
velocity of x to be Vx
x to be x
velocity of y to be Vy
y to be y

Is it supposed to look like:

function q=func(t,x,y)
q=[Vx; -x-2*x*y; Vy; -x^2+y^2]

And the solver is

clear
Vx0=0.0;
x0=0.3;
Vy0=0.0;
y0=0.0;
t=[0:0.02:200];
[x,y]=ode45(@func,Vx0,x0,Vy0,y0,[]);
plot(x,y)

I'm so confused on so many points. I don't think [x,y]=ode45(...) is right, but I don't know if it should be [t,x], [t.y] or whatever. Likewise, I know I'm meant to plot (x,y) for different time intervals, so do write plot(t,x,y)? I don't even know what [] is supposed to be for, only that it's in the notes and all the other examples I can find.

I tried so many different variations, tried to look it up in books and on the web for 7 hours straight yesterday and 2 more today and I'm feeling quite miserable about matlab. Our notes aren't helpful at all, using different variations of names for x and y for different problems which was so confusing. I'd be so appreciative for any general help on how to structure function and solver, or even direct me to a book or website that deals with two second-ODEs.

Is it just meant to look like:

function name = another function name(time, variable 1, variable 2)
name=[RHS equation 1;RHS equation 2;RHS equation 3;RHS equation 4];

And the solver:

clear
initial conditions 1 = [x];
initial conditions 2 = [y];
initial conditions 3 = [z];
constant = [a];
t = [...;...;...]
[?]=ode45(@function name,t, initial conditions 1, initial conditions2, initial conditions 3, [], constant);
plot (?)

Many thanks

Last edited:

Never mind, figured out where I went wrong.

Learning physics by trial and error...hmrph.

Never mind, figured out where I went wrong.

Learning physics by trial and error...hmrph.

thats pretty much how you learn matlab =\

thats pretty much how you learn matlab =\

Feels like how I've learned the whole of my classical mechanics module - it's not very efficient!

Hi,
I'm also trying to solve two second ODE's for a different physical quantity. Could you please let me know the right ode structure.