MATLAB: How can i get an improved Euler’s method code for this function

matlab code euler's method numerical analysis

 dy = @(x,y).2*x*y;
f = @(x).2*exp(x^2/2);
x0=1;
xn=1.5;
y=1;
h=0.1;
fprintf ('x \t \t y (euler)\t y(analytical) \n') % data table header
fprintf ('%f \t %f\t %f\n' ,x0,y,f(x0));
for x = x0 : h: xn-h
y = y + dy(x,y)*h;
x = x + h ;
fprintf (
'%f \t %f\t %f\n' ,x,y,f(x));
end

Best Answer

There are two problems with your code:
  • The analytical solution is incorrect
  • You increment x inside the for loop. Don't. The for loop does this automatically.
Here is a corrected version:
a  = 0.2;  
y0 = 1;   
x0 = 1;
xn = 1.5;
h  = 0.1;  
dy = @(x,y)a*x*y;  % dy/dx
f = @(x) y0*exp(a/2*(x.^2-1));  % Correct analytic solution
y = y0;
fprintf ('x \t \t y (euler)\t y(analytical) \n') % data table header
fprintf ('%f \t %f\t %f\n' ,x0,y,f(x0));
for x = x0+h : h: xn
  y = y + dy(x,y)*h;
  fprintf ('%f \t %f\t %f\n' ,x,y,f(x));
end
Choose a smaller step length h to for better accuracy. Alternatively try a higher order method like Runge-Kutta.
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