Coding a system of differential equations

This is the system

dx/dt=y^2/x ,

dy/dt=x^2/y

with initial conditions x(0)=√ 2 , y(0)=0

i know how to do them separately but how do you put them in a system

Answer 1

[Torsten]

Numerically solving the original system will make difficulties because of the singularity at t=0 for y.

So let's first rewrite the equations:

y^2 = x*dx/dt
-> 2*y*dy/dt = x*d^2x/dt^2 + (dx/dt)^2
-> 0.5*(x*d^2x/dt^2 + (dx/dt)^2) = x^2
-> x*d^2x/dt^2 + (dx/dt)^2 - 2*x^2 = 0

Thus solve

x*d^2x/dt^2 + (dx/dt)^2 - 2*x^2 = 0
x(0) = sqrt(2), dx/dt(0) = 0

and it follows that

y = sqrt(x*dx/dt) or y=-sqrt(x*dx/dt)

Since you said you know how to solve one equation, I think you need no further help.

[James Sm.]

So i just solve the last one and i'm done?

[Torsten]

Yes, you solve

x*d^2x/dt^2 + (dx/dt)^2 - 2*x^2 = 0
x(0) = sqrt(2), dx/dt(0) = 0

and you are done.

Because of the singularity at t=0 for y, you get two possible solutions for y that can directly be deduced from the solution for x from above:

y = sqrt(x*dx/dt) or y=-sqrt(x*dx/dt)

[James Sm.]

I'm sorry i'm fairly new to matlab and i haven't solved this kind of equation

can you help

[Torsten]

function main
  x0 = [sqrt(2);0];
  tspan = [0 2];
  fun = @(t,x)[x(2);(-x(2)^2+2*x(1)^2)/x(1)];
  [T,X] = ode15s(fun,tspan,x0);
  Y = sqrt(X(:,1).*X(:,2));   % or Y = -sqrt(X(:,1).*X(:,2));
  SOL = horzcat(X(:,1),Y);
  plot(T,SOL)
end

[James Sm.]

ok ok i somewhat get it now

thank you

[Torsten]

Remark:

The second-order ODE for x has to be rewritten as a system of two first-order ODEs where

x(1) = x and x(2) = dx/dt.

The system reads

dx(1)/dt = x(2)
dx(2)/dt = (-x(2)^2+2*x(1)^2)/x(1)

This is implemented in

fun = @(t,x)[x(2);(-x(2)^2+2*x(1)^2)/x(1)];