MATLAB lessons N.2

A simple function used to test ode45: File* f.m

:open:
 1% A comment is a line beginned with the '%' character
 2%
 3% Define the function f(x,y) = x^2 - y to be integrated
 4% by a numerical solver of MATLAB
 5%
 6% A function is defined by the keyword function
 7% followed by the prototype of the functions itself
 8
 9function res = f( x, y )
10  res = x^2 - y ;
11end

A simple function whise solution lie on a circle: File f1.m

:open:
 1%
 2%  function for the definition of the ODE system
 3%
 4%  y1'=y2
 5%  y2'=-y1
 6%  y1(0)=1
 7%  y2(0)=0
 8%
 9function res = f1( x, Y )
10  res = [ Y(2) ; -Y(1) ] ; % return a matrix with 2 row and 1 column 
11end

Another example of ODE: File f3.m

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 1%
 2% ODE:
 3%  y'=v
 4%  y'=exp(x)-v-y
 5%
 6function res = f3( x, Y )
 7  % y = Y(1)
 8  % v = Y(2)
 9  res = [ Y(2) ; exp(x) - Y(1) - Y(2) ] ;
10end

An exact solution to test the code: File exact.m

:open:
 1%
 2% Exact solution for problem
 3%
 4% y'''(t) = -t - y(t)
 5% y(0)   = y'(0) = 0
 6% y''(0) = 1
 7%
 8function res = exact( t )
 9  res = -t + (2/sqrt(3))*exp(t/2).*sin((sqrt(3)/2)*t) ;
10end