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 
and Introduce new variables: Let 
. Then, the system of first-order equations becomes:
. Then, the system of first-order equations becomes:
Set up initial conditions: From the problem, 
and 
.
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.
Hope this helps!
