solve second order differential equations in matlab

To solve second order differential equations in MATLAB, we can use the ode45 function, which is a built-in function that solves ordinary differential equations (ODEs) using a fourth and fifth order Runge-Kutta method.

The general form of a second order differential equation is:

main.m
y''(t) = f(t, y(t), y'(t))
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where y(t) is the unknown function to solve, f(t, y(t), y'(t)) is a given function, y'(t) is the derivative of y(t) with respect to t, and y''(t) is the second derivative of y(t) with respect to t.

To use ode45 to solve a second order differential equation, we need to convert the equation into a system of first order differential equations. To do this, we can define a new variable z(t) = y'(t) and rewrite the equation as follows:

main.m
y'(t) = z(t)
z'(t) = f(t, y(t), z(t))
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We can now use ode45 to solve the system of two first order differential equations. Here's an example code for solving a second order differential equation using ode45:

main.m
% Define the given function
f = @(t, y) -y - 2*y';

% Define the initial conditions
y0 = 0;
z0 = 1;
initial_conditions = [y0, z0];

% Define the time range and call ode45
tspan = [0, 10];
[t, sol] = ode45(@(t, y) [y(2); f(t, y(1), y(2))], tspan, initial_conditions);

% Plot the solution
plot(t, sol(:, 1));
xlabel('t');
ylabel('y(t)');
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In this example, the given function is -y - 2*y', the initial conditions are y(0) = 0 and y'(0) = 1, and the time range is from t = 0 to t = 10. The solution is plotted as y(t) versus t.