How to solve two differential equations using ode45.

Question

Ebraheem Menda 2018-2-22

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评论： Ebraheem Menda 2021-7-1

 My system is this
x"+ x'+ x + y'=0; 
y"+ y'+ y + x'=0; 
i need to solve these differential equations using ode's.
thanks in advance.

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Torsten 2018-2-22

You will need four initial conditions (for x,x',y,y') if you want to use ODE45 as numerical solver.

Ebraheem Menda 2018-2-22

Thank you Torsten. i have the initial conditions. but my question is how to convey these equations to ode45 or any other solver. Because they are coupled equations. thanks for your help.

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Answer 1

Torsten 2018-2-22

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https://ww2.mathworks.cn/matlabcentral/answers/384186-how-to-solve-two-differential-equations-using-ode45#answer_306512

在 MATLAB Online 中打开

fun=@(t,y)[y(2);-y(2)-y(1)-y(4);y(4);-y(4)-y(3)-y(2)];
x0=...;      %supply x(t0)
x0prime=...; %supply x'(t0)
y0=...;      %supply y(t0)
y0prime=...; %supply y'(t0)
Y0=[x0; x0prime; y0; y0prime];
tspan=[0 5];
[T,Y]=ode45(fun,tspan,Y0);
plot(T,Y(:,1))  %plot x
plot(T,Y(:,3))  %plot y

Best wishes

Torsten.

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Andreas Zimmermann 2020-4-16

编辑：Andreas Zimmermann 2020-4-16

在 MATLAB Online 中打开

Hi Joe,

Using Torsten's relations

x = y(1)

x' = y(2)

y = y(3)

y' = y(4)

the first line

fun=@(t,y)[y(2); -y(2)-y(1)-y(4); y(4); -y(4)-y(3)-y(2)];

can be translated to the following, using Torsten's relations:

[x'; -x'-x-y'; y'; -y'-y-x']

which is essentially

[x'; x"; y'; y"].

He got these relations by solving for x" and y":

x"+ x'+ x + y'=0; => x" = -x' -x -y' = -y(2) -y(1) -y(4)

y"+ y'+ y + x'=0; => y" = -y' -y +x' = -y(4) -y(3) -y(2)

I have tried to use ode45 to solve the SIR model from this fantastic Geogebra Tutorial: (https://youtu.be/k6nLfCbAzgo)

% Just some start parameters and coefficients
Istart  = 0.01;
Sstart  = 0.99;
Rstart  = 0;
transm  = 3.2;
recov   = 0.23;
maxT    = 20;
% define S',I',R'
% S' = - transm * S * I
% I' = transm * S I - recov * I
% R' = recov * I
% S is y(1)
% I is y(2)
% R is y(3)
%so here fun = @(t,y)[S'; I'; R'];
fun     = @(t,y)[-transm*y(1)*y(2); (transm*y(1)*y(2))-(recov*y(2)); recov*y(2)];
% Provide the starting parameters
Y0      = [Sstart; Istart; Rstart;];
% Define the range of t
tspan   = [0 maxT];
% Magic happens and matrix Y contains S,I,R
[T,Y]   = ode45(fun,tspan,Y0);
% Plot plot
figure
plot(T,abs(Y(:,1)),'b-') % S
hold on
plot(T,abs(Y(:,2)),'r-') % I
plot(T,abs(Y(:,3)),'g-') % R
xlim([0 23])
%Hope this helps :)
% Many thanks to Torsten and Ebraheem for your very very useful discussion!

Ebraheem Menda 2020-7-19

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Answer 2

Abraham Boayue 2018-2-24

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Hey Ebraheem There are many excellent methods that you can use to solve your problem, for instance, the finite difference method is a very powerful method to use. I can try with that.The ode45 function is a matlab built in function and was designed to solve certain ode problems, it may not be suitable for a number of problems. What are the initial values of your equations? Do you have any plot of the solution that one can use as a guide?

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Ebraheem Menda 2018-2-24

thank you Abraham for your response. i am yet to solve those equations and obtain initial conditions. once i got them i will post the equations. Meanwhile what is this finite difference method ? is it available in matlab 2009b ?

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Answer 3

Abraham Boayue 2018-2-24

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The finite difference method is used to solve differential and partial equations. It is easier to implement in matlab. You can do the coding in any version of matlab, I have taken a course in numerical mathematics before and have a fairly good knowledge of how to solve such problems.