ode second order in matlab

To solve a second order differential equation in MATLAB, we can use the built-in function ode45(). This function uses an iterative method to solve initial value problems (IVP) of the form:

main.m
y'' = f(t, y, y')
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where t is the independent variable (often time), y is the dependent variable, y' and y'' are its first and second derivatives, and f(t, y, y') is the right-hand side of the differential equation.

To use ode45() for a second order problem, we need to convert it into a system of first order differential equations, by introducing a new variable z:

main.m
y' = z
z' = f(t, y, z)
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We can then use ode45() to solve this system of first order equations.

Here's an example of using ode45() to solve the second order differential equation y'' + 2y' + y = 0, with initial conditions y(0) = 0 and y'(0) = 1:

main.m
% Define the right-hand side of the differential equation
f = @(t, y) [-2*y(2) - y(1)];

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

% Define the time interval to solve for
tspan = [0, 10];

% Solve the differential equation
[t, y] = ode45(f, tspan, y0);

% Plot the result
plot(t, y(:,1), '-o');
xlabel('time t');
ylabel('y(t)');
title('Solution of y'''' + 2y'' + y = 0');
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This code defines the function f as -2*y(2) - y(1), sets the initial conditions y0 to [0, 1], and the time interval tspan to [0, 10]. It then calls ode45(f, tspan, y0) to solve the differential equation, and finally plots the solution.