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Please help solve a system of differential equation

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I have a system of differential equation in x,y with boundary conditions. I want to solve it analytically and numerically. The analytical solutions coming in the form of error functions. Can someone please help me find analytical and numerical solution for this, I have been trying for weeks now, really need to solve this for my thesis. Please help.

Answers (1)

Alan Stevens
Alan Stevens on 15 Mar 2023
Here's a numerical approach:
First manipulate the equations
tspan = [0, 120];
ic = [80000, 0.001];
[t, xy] = ode45(@rate, tspan, ic);
x = xy(:,1); y = xy(:,2);
subplot(2,1,1)
plot(t,x),grid
xlabel('t'), ylabel('x')
subplot(2,1,2)
plot(t,y),grid
xlabel('t'), ylabel('y')
function dxydt = rate(~,xy)
a = 0.1; b = 0.2; c = -0.1; % Replace with desired values
x = xy(1); y = xy(2);
dxydt = [a-b+c*x*y; % eqn (3)
c*y*(1-y) - a*y/x]; % eqn (5)
end
  19 Comments
Jayesh Jawandhia
Jayesh Jawandhia on 4 Apr 2023
Hope you are doing well. I have been reading up on these and your solution has been really helpful. I just had a follow-up. We are trying to find a better fit by making a and b as a function of t. I haven't been able to figure out how do I add these to my ode45 solver in Matlab just like what you did here.
Can you please help with this? For simplicity, I am assuming both a and b second order in t, i.e.:
a = mt^2+nt+r
b = pt^2+qt+s
Really thankful for all your help!
Torsten
Torsten on 4 Apr 2023
What's the problem ?
function dxydt = rate(t,xy);
m = ...;
n = ...;
r = ...;
p = ...;
q = ...;
s = ...;
a = m*t^2+n*t+r;
b = p*t^2+q*t+s;
x = xy(1);
y = xy(2);
dxydt = [a-b+c*x*y;c*y*(1-y)-a*y/x];
end

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