Derivation of the ODE's from the original PDE's.

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Here is the boundary-layer momentum equation

Now we make the subsitutions for the velocities,

This step replaces derivatives of η.

Now that we are done taking derivatives we can make the η(x,y) into just η.

We now clean it all up and remove common factors,

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The final boundary-Layer ODE is

Boundary conditions

The boundary conditions are that there is no slip on the surface, no flow through the surface and that the tangential velocity matches the free stream far away.

The are in terms of the newest notation:

u(x,y=0)=0, η = 0, f'(η= 0)=0,
v(x,y=0)=0, η = 0, f(η= 0=0,
y->∞, u(x,y->∞)=U, f'(η=∞) = 1,  
x=0, u(x,y->∞)=U, f'(η=∞) = 1,