Example 4.4-2

Boundary Layer "Theory" applied to flow near the Leading edge of a flat Plate.
>  restart;
>  econ:=D[1](vx)(x,y)+D[2](vy)(x,y)=0; Continuity Equation
>  emotx:=(x,y)->vx(x,y)*D[1](vx)(x,y)+vy(x,y)*D[2](vx)(x,y)-
nu*D[2](D[2](vx))(x,y); x component of eq. of motion
>  vy:=(x,y)->-int(D[1](vx)(x,y1),y1=0...y); This should be a 
solution to the continuity equation.
>  econ; It is!
>  emotx(x,y); This gives 4.4-14
>  assume(vinf>0);
>  vx:=(x,y)->vinf*phi(y/delta(x)); The assumed form of solution 
suggested in eq. 4.4-15
Here is eq. 14 after this substitution.
>  emotx(x,y);
>  eq19:=int(emotx(x,y),y=0...delta(x)); Now integrate wrt y 
from 0 to delta(x).
>  phi:=eta->(3/2)*eta-(1/2)*eta^3; One possible polynomial that 
vanishes at eta=0 and has a zero derivative at eta = 1. Lots of 
others could be used.
>  eq19; This is a DE that determines the variation of delta with x.
>  s:=dsolve({eq19=0,delta(0)=0},delta(x)); Solving for delta 
and using the BC that it should be zero at x=0.
>  assign(s[2]); This checks with eq. 4.4-25
>  simplify(evalf(delta(x)));  Thus if 
phi(eta)=(3/2)*eta-(1/2)*eta^3, delta(x) is: