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How to make a phase diagram plot?

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Wing Nam
Wing Nam 2023-4-1
回答： Sam Chak 2023-12-16
Hi all. I have an ODE and I have already found the general solution of.
How can I plot a phase diagram with some initial value like y(1)=1?
  1 个评论
Sai
Sai 2023-4-4
I am not sure if the solution is correct because when x is equal to 1, y is not equal 1.

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回答（2 个）

Sai
Sai 2023-4-4
Hi Wing,
I understand that you want to make a phase diagram plot for the above differential equation.
Please refer to the below code snippet.
function dydx = diff_eq(x,y)
dydx = [y(2); (exp(-x)-2*(x-1)*y(2)-(x-2)*y(1))/x];
end
Place two code snippets in different .m files. But the above code snippet in diff_eq.m file
[x,y] = ode45(@diff_eq,[1 5],[1; 1]);
plot(y(:,1),y(:,2));
xlabel('y(1)');
ylabel('y(2)');
please refer to the below documentation ink for more information on using ‘ode45’
I hope this helps you to resolve the query
Sam Chak
Sam Chak 2023-12-16
Ran in:
Hi @Wing Nam
If the analytical solution exists, then you can solve the ODE symbolically using the dsolve() command. From the results, you can plot out the phase portrait diagram using the fplot() command.
%% Setup the ODE in symbolic form
syms y(x) z(x)
Dy = diff(y,x); % dy/dx
ODEqn = diff(y,x,2) == (exp(-x) - 2*(x - 1)*Dy - (x - 2)*y)/x
ODEqn(x) = 
%% General solution
yGenSol(x) = dsolve(ODEqn)
yGenSol(x) = 
%% Particular solution
init = [y(1)==1, Dy(1)==0]; % initial values
yParSol(x) = dsolve(ODEqn, init)
yParSol(x) = 
DyParSol = simplify(diff(yParSol), 'steps', 100)
DyParSol(x) = 
%% Plot time responses of states
figure(1)
T1 = tiledlayout(2, 1);
nexttile(T1)
fplot( yParSol, [1 20]), grid on, ylabel('y(x)')
nexttile(T1)
fplot(DyParSol, [1 20]), grid on, ylabel('dy/dx')
title( T1, '2nd-order System')
xlabel(T1, 'Time')
ylabel(T1, 'Staes')
%% Plot phase portrait diagram
figure(2)
fplot(yParSol, DyParSol, [1 20]), grid on
title('Phase Portrait')
xlabel('y(x)'), ylabel('dy/dx')

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