Problem
Write a function to solve for u and v as a function of z in this system of ordinary differential equations:
where f, U, V, and η are constants. The boundary conditions are that 
at 
and 
and 
as 
.
at and and as .Background
This set of equations results from simplifying the Navier-Stokes equations (i.e., conservation of momentum for a fluid with a linear stress-rate of strain relation) for large-scale flow subjected to rotation. The horizontal velocity components are u and v. The Coriolis parameter f is related to Earth’s rotation rate and the latitude, and η is the kinematic viscosity, which can be interpreted as an eddy viscosity to account for the effects of turbulence.
The velocities far above the surface, U and V, result from the pressure gradient: 
and 
, where p is pressure and ρ is density. Notice that far above the surface, the flow is along the isobars, or lines of constant pressure. As the surface is approached, the velocity vector rotates—or spirals.
and , where p is pressure and ρ is density. Notice that far above the surface, the flow is along the isobars, or lines of constant pressure. As the surface is approached, the velocity vector rotates—or spirals. Solution Stats
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"where f, U, V, and v are constants" - do you mean f, U, V and nu?
To reduce confusion, I changed nu (which Cody rendered closer to v) to eta.