I'm trying to solve a 2nd order ode with ode45, but have no idea where to start.

This the ode with conditions I'm trying to solve and the code below is as far as I got. It would be appreciated if I could get a detailed step by step to help solve this.

%initial conditions
y0 = [0 1];
tspan = [1 4];

Answer 1

[Milan Bansal]

Hi Jhryssa,

Here is how you can solve the given ODE.

Convert the second-order ODE into a system of first-order ODEs: MATLAB's ode45 solver is designed for systems of first-order ODEs, so you will need to rewrite the second-order ODE into a system of two first-order equations.

The given ODE is:

with the initial conditions and

Introduce new variables: Let . Then, the system of first-order equations becomes:

Set up initial conditions: From the problem, and .

Here is the MATLAB implementaion:

% Define the system of first-order ODEs
function dydt = ode_system(t, y)
    dydt = zeros(2,1);
    dydt(1) = y(2);
    dydt(2) = (2*y(1) - t^2*y(2)) / t;
end

% Set initial conditions
y0 = [0; 1];  % y(1) = 0, y'(1) = 1

% Time span (t = 1 to t = 4)
tspan = [1 4];

% Solve the ODE using ode45
[t, y] = ode45(@ode_system, tspan, y0);

% Plot the result
plot(t, y(:,1))
xlabel('t')
ylabel('y(t)')
title('Solution of the ODE')

Please refer to the documentation link to learn more about ode45.

https://www.mathworks.com/help/matlab/ref/ode45.html

Hope this helps!

[Jhryssa]

Thank you so much! I understand how to do this a lot more now!

Answer 2

[Torsten]

After dividing your equation by t, you can just follow the example

Solve Nonstiff Equation

under

https://uk.mathworks.com/help/matlab/ref/ode45.html