# ODE for 2 variables

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Nivedhitha S on 6 Aug 2019
Commented: Torsten on 19 Aug 2019
Hello all,
Does matlab support DE of this type: d(xy)/dt (ie. y dx/dt + x dy/dt)? If so how to werite these type of equations? I am a new comer to matlab and any help would be great!
Thanks.

Torsten on 6 Aug 2019
You will need two equations to determine x and y. What is your second equation ?
Nivedhitha S on 6 Aug 2019
I have set of 4 interdependent eqns of the type d(xy)/dt, say something like this, where p1-p7 are parameters:
1. dx/dt=A;
2. d(xy)/dt=A*(p1-y)+p2*x*y
3. d(xz)/dt= A*(p3-z)-p4*x*y
4. d(xu)/dt= p5*x*(p6-u)-p7*x*y

Torsten on 6 Aug 2019
Edited: Torsten on 6 Aug 2019
function main
p = [1 2 3 4 5 6 7];
A = 20;
x0 = 2;
y0 = 5;
z0 = 27;
u0 = -34;
a0 = [x0; x0*y0; x0*z0; x0*u0];
tspan = [0, 1];
[T a] = ode45(@(t,a)fun(t,a,A,p),tspan,a0)
end
function da = fun(t,a,A,p)
x = a(1);
y = a(2)/a(1);
z = a(3)/a(1);
u = a(4)/a(1);
da = zeros(4,1);
da(1) = A;
da(2) = A*(p(1)-y)+p(2)*a(2);
da(3) = A*(p(3)-z)-p(4)*a(2);
da(4) = p(5)*x*(p(6)-u)-p(7)*a(2);
end

Nivedhitha S on 17 Aug 2019
That was very useful! Thanks a lot!
Can I creat few equantions which has to be integrated in a different time range within the same function?
say
da (5) = p(6)*x*y;
but this has to be active in a tspan of (t2:t3) whereas the other 4 has to be active in the tspan of (t1:t3). Will this be possible to do?
Torsten on 19 Aug 2019
You will have to call the ODE integrator two times (from t1 to t2 and from t2 to t3) and adjust the equations to be integrated depending on the time span (4 active equations from t1 to t2 and 5 active equations from t2 to t3).

Steven Lord on 6 Aug 2019
Write your equations in the form M*DV = RHS where: , and M is the mass matrix. Since your second equation expands to: the second row of your mass matrix will be [y, x, 0, 0]. Multiply that vector times DV and you'll see that you've recreated the left side of the second differential equation. Generate the remaining rows of the mass matrix similarly.
Create an options structure that specifies the mass matrix using odeset and the 'Mass' name-value pair. The value for that pair will be a function handle that accepts t (time) and the vector V and returns the mass matrix M.
Then write the function that evaluates RHS as a function of t and V. Call the ODE solver specifying that function and the options structure (so the solver knows how to create the mass matrix.