How to make a phase diagram plot?

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?

[Sai]

I am not sure if the solution is correct because when x is equal to 1, y is not equal 1.

Answer 1

[Sai]

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’

• https://in.mathworks.com/help/matlab/ref/ode45.html#:~:text=The%20van%20der,%27y_2%27)

I hope this helps you to resolve the query

Answer 2

[Sam Chak]

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')