Help in solving the following bvp

Question

  eqn2 = -dp/L + \[Mu]*D[v[x, y], x, x] - 
        Subscript[\[Sigma], e]*Subscript[B, x]^2*v[x, y] + 
        Subscript[\[Sigma], e]*Subscript[E, z]*Subscript[B, x] + \[Zeta] Cosh[x \[Kappa]] Sech[\[Kappa]]*
         Subscript[E, y] == 0;
    bc2 = { v[a, y] == 0, Derivative[0, 1][v][0, y] == 0};
    sol2 = DSolveValue[{eqn2, bc2}, v[x, y], {x, y}];
    Assuming[{a > 0, \[Kappa] > 0}, Simplify[sol2]]

I am not able to get the solution

Answer 1

Hint.

Try to solve first

eqn2 = -dp/L + \[Mu]*D[v[x, y], x, x] - 
    Subscript[\[Sigma], e]*Subscript[B, x0]^2*v[x, y] + 
    Subscript[\[Sigma], e]*Subscript[E, z]*
    Subscript[B, x0] + \[Zeta] Cosh[x \[Kappa]] Sech[\[Kappa]]*
    Subscript[E, y0] == 0;
bc2 = {v[a, y] == 0 (***,Derivative[0,1][v][0,y]==0 ***)};
sol2 = DSolveValue[{eqn2, bc2}, v[x, y], {x, y}] // FullSimplify

Note that now we are using Subscript[B, x0] and Subscript[E, y0] instead of Subscript[B, x] and Subscript[E, y] and also considering only the first boundary condition. The second boundary condition can be handled after that imposing

D[(sol2/.{x -> 0}), y] == 0