First of all ode45 accepts functions with two arguments t - e.g. time and y - e.g. position, velocity, etc..

Functions should return the derivative.

It means that ode45 solves equations in form: y' = f(t,y)

For example if you want to solve equation:

dy/dt = sin(t) + y

with initial condition y(0)=0

(the analytical solution is: y(t) = 1/2*(e^t-sin(t)-cos(t)))

The numerical solution

ode45(@(t,y)sin(t)+y,[0,1],0)

hold all

% comparison with the analytical solution

ezplot('1/2*(exp(t)-sin(t)-cos(t))',[0 1])

More complex example:

differential equation:

y''' - 2y'' - y' + 2y = 0

initial conditions:

y(0) = 0, y'(0) = 0, y''(0) = 1

the analytical solution:

y(t) = 1/6*e^(-t)*(e^t-1)^2*(2*e^t+1)

numerical solution:

dy = @(t,y)[y(2);y(3);2*y(3)+y(2)-2*y(1)];

ode45(dy,[0,1],[0;0;1]);

hold all

% comparison with the analytical solution

ezplot('1/6*exp(-t)*(exp(t)-1)^2*(2*exp(t)+1)',[0 1])

Anonymous function dy can be replaced by standard MATLAB function:

function F = dy(t,y)

% y''' = 2y'' + y' - 2y

% y(1) = y

% y(2) = y'

% y(3) = y''

F(1,1) = y(2);%y'

F(2,1) = y(3);%y''

F(3,1) = 2*y(3)+y(2)-2*y(1);%y'''

Grzegorz

thank you so much! it was a nice contribution of yours. it helped me find a very silly mistake of mine!!!