ODE solvers (ode45) in MATLAB

Introduction

ODE solvers like ode45 help you find solutions to equations that describe how things change over time. They do the hard math for you.

When you want to model how a population grows or shrinks over time.

When you need to simulate the motion of a falling object with air resistance.

When you want to understand how a chemical reaction changes concentration over time.

When you want to predict the temperature change in a cooling cup of coffee.

When you want to solve physics problems involving changing speeds or positions.

Syntax

MATLAB

tspan = [t0 tf];
[t, y] = ode45(@odefun, tspan, y0);

function dydt = odefun(t, y)
    dydt = ...; % define the derivative here
end

tspan is the time interval you want to solve over, from start t0 to end tf.

odefun is a function that returns the rate of change (derivative) at time t and state y.

Examples

This solves the simple decay equation dy/dt = -2y from time 0 to 5 starting at y=1.

MATLAB

tspan = [0 5];
y0 = 1;
[t, y] = ode45(@(t,y) -2*y, tspan, y0);

This solves a pendulum motion where y(1) is angle and y(2) is angular velocity.

MATLAB

function dydt = pendulum(t, y)
    g = 9.81; L = 1;
    dydt = [y(2); -(g/L)*sin(y(1))];
end

[t, y] = ode45(@pendulum, [0 10], [pi/4; 0]);

Sample Program

This program solves the equation dy/dt = -2y and shows the values of y at different times.

MATLAB

function simple_decay_example
    tspan = [0 5];
    y0 = 1;
    [t, y] = ode45(@(t,y) -2*y, tspan, y0);
    disp(table(t, y))
end

Important Notes

ode45 is good for many problems but not all. It works best for smooth problems without sudden jumps.

You can plot the results using plot(t, y) to see how the solution changes over time.

Summary

ode45 solves equations that describe change over time.

You give it a function for the rate of change, a time range, and a starting value.

It returns time points and the solution values at those points.