Solve this mixed partial differential equation

Questions
1

This is my QS:

Mixedpartialdifferentialequation.png

This is my Code:

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2

4438Bez tytułu.png
4438Bez tytułu.png857×159 7.54 KB

So you have the system and is underdetermined. That means the indicated function does not include enough equations to specify a solution.

3

Thanks

4

I add more equation as:

6818Mixedpartialdifferentialequation.png

This is my code

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It is still not solvable.

5

I tried with DSolve and NDSolve, but Mathematica fails.

With Maple 2020 I have:

4519Bez tytułu.png
4519Bez tytułu.png1200×333 16.3 KB

Translating to Mathematica code:

    ClearAll["`*"]
    sol = {R[z] -> R0*Exp[((Log[U0/UL] + 2*I*Pi*n)*z)/(2 L)], 
      u[z] -> U0/
       Exp[((Log[U0/UL] + 2*I*Pi*n)*z)/(2 L)]^2};(*  n\[Element]Integers *)
    
   u[z] /. sol[[2]] /. 
    z -> 0(*The equation satisfy a initial condition u[0]=U0 *)

    (* U0 *)

    {sol[[1]] /. n -> 0, sol[[2]] /. n -> 0}(*for n=0*)
    
    (* {R[z] -> R0 (U0/UL)^(z/(2 L)), u[z] -> U0 (U0/UL)^(-(z/L))} *)
6

Your second equation is not a differential equation. You can remedy by taking its derivative:

eq = 2*D[r[z], z]*D[u[z], z] +
   r[z]*D[D[u[z], z], z] == 0
sol = DSolve[{eq,
   D[r[z]^2*u[z] == r0^2*u0, z],
   r[0]^2*u[0] == r0^2*u0},
  {u, r}, z]

A trivial example that Mathematica does not solve:

DSolve[{a'[x] == 0, b[x] == a[x]^2}, {a, b}, x]