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Solving 3 simultaneous differential equations

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Project Description

I have a system of 3 simulatenous differential equations (equations 1,2 and 3) that I need to solve in ur(r), ut(r) and phi(r). These are engineering equations so the solution has to make sense.
These equations are simplified using physics to two equations (equations 4 and 5) in ur(r) and phi(r)
Project tasks:
Verify my derivation for the equations
Solving the differential equations (equations 1,2 and 3) simulatenously...in closed form if possible if not in iterative
Solving the differential equations (equations 4 and 5) simulatenously...I think they can be solved in closed form solution since I can find solution to each but not simultaneously...
See attached file

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