# I am trying to solve 3 simultaneous ODE's. Which method to use?

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Dhiraj Deshmukh on 6 Sep 2021
Commented: Dhiraj Deshmukh on 7 Sep 2021
I am trying to solve 3 simultaneous ODE's. Which method should be used to solve these equations?? I have tried to solve it using ode45 method but I'm unable to get the answer.
dx1/df = F( f, x1, x2, x3 ) ]
dx2/df = F( f, x1, x2, x3 )
dx3/df = F( f, x1, x2, x3 )
f = [0.5 1]
x1(0.5)=0.185, x2(0.5)= 0.285, x3(0.5)= 0.53
gamma = 0.577;
xo1= 0.185; xo2= 0.285; xo3 =0.53;
theta=0.5;
q1=1; q2 = 0.317; q3 = 0.065;
F(x1,f) = ((q1)*(x1-gamma*(x1*f-(xo1*(1-theta)))/(f-(1-theta)))-x1*(x1-gamma*(x1*f-(xo1*(1-theta)))/(f-(1-theta))+q2*(x2-gamma*(x2*f-(xo2*(1-theta)))/(f-(1-theta)))+q3*(x3-gamma*(x3*f-(xo3*(1-theta)))/(f-(1-theta))))*f)/(((x1-gamma*(x1*f-(xo1*(1-theta)))/(f-(1-theta)))+q2*(x2-gamma*(x2*f-(xo2*(1-theta)))/(f-(1-theta)))+q3*(x3-gamma*(x3*f-(xo3*(1-theta)))/(f-(1-theta))))*f);
F(x2,f) = ((q2)*(x2-gamma*(x2*f-(xo2*(1-theta)))/(f-(1-theta)))-x2*(x1-gamma*(x1*f-(xo1*(1-theta)))/(f-(1-theta))+q2*(x2-gamma*(x2*f-(xo2*(1-theta)))/(f-(1-theta)))+q3*(x3-gamma*(x3*f-(xo3*(1-theta)))/(f-(1-theta))))*f)/(((x1-gamma*(x1*f-(xo1*(1-theta)))/(f-(1-theta)))+q2*(x2-gamma*(x2*f-(xo2*(1-theta)))/(f-(1-theta)))+q3*(x3-gamma*(x3*f-(xo3*(1-theta)))/(f-(1-theta))))*f);
F(x3,f)= ((q3)*(x3-gamma*(x3*f-(xo3*(1-theta)))/(f-(1-theta)))-x3*(x1-gamma*(x1*f-(xo1*(1-theta)))/(f-(1-theta))+q2*(x2-gamma*(x2*f-(xo2*(1-theta)))/(f-(1-theta)))+q3*(x3-gamma*(x3*f-(xo3*(1-theta)))/(f-(1-theta))))*f)/(((x1-gamma*(x1*f-(xo1*(1-theta)))/(f-(1-theta)))+q2*(x2-gamma*(x2*f-(xo2*(1-theta)))/(f-(1-theta)))+q3*(x3-gamma*(x3*f-(xo3*(1-theta)))/(f-(1-theta))))*f);
Dhiraj Deshmukh on 6 Sep 2021
code i have used :-
function [dy] = countermulticompfcn(f,y)
theta = 0.5;
xo1 = 0.185;
xo2 = 0.285;
xo3 = 0.53;
gamma = 0.577;
q1=1;
q2 = 0.3543;
q3 = 0.3249;
dy = zeros(4,1)
dy(1) = ((q1)*(y(1)-gamma*(y(1)*f-(xo1*(1-theta)))/(f-(1-theta)))-y(1)*(y(1)-gamma*(y(1)*f-(xo1*(1-theta)))/(f-(1-theta))+q2*(y(2)-gamma*(y(2)*f-(xo2*(1-theta)))/(f-(1-theta)))+q3*(y(3)-gamma*(y(3)*f-(xo3*(1-theta)))/(f-(1-theta)))))/(((y(1)-gamma*(y(1)*f-(xo1*(1-theta)))/(f-(1-theta)))+q2*(y(2)-gamma*(y(2)*f-(xo2*(1-theta)))/(f-(1-theta)))+q3*(y(3)-gamma*(y(3)*f-(xo3*(1-theta)))/(f-(1-theta))))*f)
dy(2) = ((q2)*(y(2)-gamma*(y(2)*f-(xo2*(1-theta)))/(f-(1-theta)))-y(2)*(y(1)-gamma*(y(1)*f-(xo1*(1-theta)))/(f-(1-theta))+q2*(y(2)-gamma*(y(2)*f-(xo2*(1-theta)))/(f-(1-theta)))+q3*(y(3)-gamma*(y(3)*f-(xo3*(1-theta)))/(f-(1-theta)))))/(((y(1)-gamma*(y(1)*f-(xo1*(1-theta)))/(f-(1-theta)))+q2*(y(2)-gamma*(y(2)*f-(xo2*(1-theta)))/(f-(1-theta)))+q3*(y(3)-gamma*(y(3)*f-(xo3*(1-theta)))/(f-(1-theta))))*f)
dy(3) = ((q3)*(y(3)-gamma*(y(3)*f-(xo3*(1-theta)))/(f-(1-theta)))-y(3)*(y(1)-gamma*(y(1)*f-(xo1*(1-theta)))/(f-(1-theta))+q2*(y(2)-gamma*(y(2)*f-(xo2*(1-theta)))/(f-(1-theta)))+q3*(y(3)-gamma*(y(3)*f-(xo3*(1-theta)))/(f-(1-theta)))))/(((y(1)-gamma*(y(1)*f-(xo1*(1-theta)))/(f-(1-theta)))+q2*(y(2)-gamma*(y(2)*f-(xo2*(1-theta)))/(f-(1-theta)))+q3*(y(3)-gamma*(y(3)*f-(xo3*(1-theta)))/(f-(1-theta))))*f)
end
solver:-
clear all
for index= 1:4
myprompt2=['Enter the initial value of Y:(',num2str(index),'):'];
initialval(index) = input(myprompt2);
end
lower = input('Enter Lower limit of Integration ');
upper = input('Enter Upper limit of Integration ');
[F,y] = ode45(@countermulticompfcn,[lower upper],initialval)
plot(F,y)
hold on

Paul on 6 Sep 2021
This line:
dy = zeros(4,1);
Needs to be replaced with:
dy = zeros(3,1);
After that, note that the equation for dy(1) evaluates to NaN at f = 0.5 and to -Inf at f = 0 (I only explored between 0 and 1). In the original question it was stated that the the tspan input to ode45 is [0.5 1]. So that will cause a problem because the initial value of dy is NaN at f = 0.5 and that kills the entire solution. I ran from [0.51 1] and got a non-NaN result. In the follow up comment the user enters the tspan limits. If those limits cause the solver to hit either f = 0 or f = 0.5 exactly there will be a problem. So I guess the real question is if it's expected that the equations for dy have these singulariites and how they should be handled if the solver has to integrate through them.
Dhiraj Deshmukh on 7 Sep 2021
Thank you very much Paul. Problem is solved. Your suggestions really hepled.

Steven Lord on 6 Sep 2021
This documentation page includes two examples, the second of which you can use as a model for your code to solve a system of differential equations.
Dhiraj Deshmukh on 7 Sep 2021
Thank you very much for guiding.