Numerical Solution

The primes (') denote derivatives with respect to the similarity variable.  This nonlinear ode does not have a known analytical solution.  However, it can be easily solved numerically.  
        
Numerical solutions for ode's are often done by creating a system of first order odes.  This is done by defining
        
        y1 = f,
        y2 = f'
        y3 = f'',
        
These have to be related.
        
        Thus we have
        y1' = y2 (by definition)
        y2' = y3 (by definition)
        y3' = -1/2*y1*y3   (which is from the original ode.)
        
In the first part I solve the flat plate problem separately using a crude numerical scheme that usually converges, although not real fast.  It is a shooting method.  I have picked y =15 as the end of the integrate.  The NDSolve routine uses a Runge-Kutta method with built in step size adjustment.