How to solve 4th order Runge-Kutta for different initial conditions?

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Kenny Gee
Kenny Gee on 22 Apr 2022
Commented: Kenny Gee on 24 Apr 2022
I have a code that solves the 2 populations with 1 initial conditions and just plot that. How do I make 2D plots for different initial conditions?
%dx/dt = 5*x+1*x*y
%dy/dt = -2*y+1*x*y
clear
clc
f=@(t,z) [5*z(1)+1*z(1)*z(2);
-2*z(2)+1*z(1)*z(2)];
% Initial Conditions
t(1)=0;
z(:,1)=[1,1];
% Step size
s=1;
e=10;
N=e/s;
% Update Loop
for i=1:N
% Update time
t(i+1)=t(i)+s;
% Update for y
k1=f(t(i) ,z(:,i));
k2=f(t(i)+s/2,z(:,i)+s/2*k1);
k3=f(t(i)+s/2,z(:,i)+s/2*k2);
k4=f(t(i)+s ,z(:,i)+s *k3);
z(:,i+1)=z(:,i)+h/6*(k1 + 2*k2 + 2*k3 + k4);
end
  1 Comment
Davide Masiello
Davide Masiello on 22 Apr 2022
Your code does not run because h is not defined.
Moreover, what exactly would you like to plot in 2D?

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Accepted Answer

Alan Stevens
Alan Stevens on 22 Apr 2022
Here's a rather crude method (together with some corrections). You should be able to turn this into a much more elegant version:
%dx/dt = 5*x+1*x*y
%dy/dt = -2*y+1*x*y
f=@(t,z) [5*z(1)+1*z(1)*z(2);
-2*z(2)+1*z(1)*z(2)];
% Example initial conditions
x0 = [1, 0.5, 0.1];
y0 = [-1, -0.5, -0.1];
% Loop through different initial conditions
for j = 1:numel(x0)
% Initial Conditions
z(:,1)=[x0(j),y0(j)]; %%% Make y(0) negative
% Step size
s=0.1; %%% Need much smaller step size
e=10;
N=e/s;
t = zeros(1,N);
% Update Loop
for i=1:N
% Update time
t(i+1)=t(i)+s;
% Update for y
k1=f(t(i) ,z(:,i));
k2=f(t(i)+s/2,z(:,i)+s/2*k1);
k3=f(t(i)+s/2,z(:,i)+s/2*k2);
k4=f(t(i)+s ,z(:,i)+s *k3);
z(:,i+1)=z(:,i)+s/6*(k1 + 2*k2 + 2*k3 + k4); %%% s not h
end
x = z(1,:); y = z(2,:);
figure
subplot(2,1,1)
plot(t,x),grid
xlabel('t'),ylabel('x')
subplot(2,1,2)
plot(t,y),grid
xlabel('t'),ylabel('y')
end
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