how do you solve $y''+2y'-3y=0$? I want to solve this equation:
$y''+2y'-3y=0$
I did this:
$y' = z$ 
$y'' = z\dfrac{dz}{dy}$
$z\dfrac{dz}{dy}+2z-3y=0$
$zdz+2zdy-3ydy=0$
$zdz=(3y-2z)dy$
$z=3y-2z$
$z=y$
$y=y'=y''$
???
now, I'm pretty sure I did something wrong. could you please correct.

HINT $\rm\ \ \ 0\  =\  y'' + 2\ y' - 3\ y\ =\  (D^2 + 2\ D -3)\ y\ =\ (D+3)\ (D-1)\ y, \quad D = d/dx $  
i.e. factor the differential operator as you would any polynomial.
