Matlab Code for solving and plotting function x'(t) = 1 + t*sin(t*x)

[bdoherty1994]

Hi everyone,

I am trying to solve and plot the function, x'(t) = 1 + t*sin(t*x) where x(0) = 0 and t_final = 1, in order to compare this exact solution to the approximations of Euler's and Improved Euler's Method. Can anyone help me with the code in order to solve this problem, and then plot it, using Matlab?

Thanks in advance!

 

[bigfooted]

Sure. Just show us the code that you have right now and the problems you are facing and we'll help.

 

[bdoherty1994]

syms x y
init = 'y(0)=0'
dsolve('Dy = 1 + t.*sin(t*y)', 'y(0)=0', 't')

I keep coming up with an error.

 

[the_wolfman]

There are a number of things

You define
sym x y

but use x and y in your equation

You should define
sym y(t)

not
sym y t

In your desolve statement you should use '==' not '='

Lastly I don't know if the ode has an explicit solution. If it does not, then MATLAB will still give you an warning.

 

[alphy]

Hello there,

Here is one possible solution using Matlab:

% Define the function x'(t) = 1 + t*sin(t*x)
f = @(t,x) 1 + t*sin(t*x);

% Set initial conditions and time interval
x0 = 0; % x(0) = 0
tspan = [0 1]; % t_final = 1

% Solve using Euler's method
[t_euler, x_euler] = euler(f,tspan,x0);

% Solve using Improved Euler's method
[t_improved, x_improved] = improved_euler(f,tspan,x0);

% Solve using exact solution
t_exact = linspace(0,1,100);
x_exact = (exp(t_exact) - 1) ./ t_exact; % Exact solution derived from integrating x'(t)

% Plot the results
plot(t_euler,x_euler,'r',t_improved,x_improved,'b',t_exact,x_exact,'k--')
legend('Euler method','Improved Euler method','Exact solution')
xlabel('t')
ylabel('x(t)')

I hope this helps! Let me know if you have any further questions or if you need help understanding any part of the code. Keep up the good work with your project!