have formulated a problem in which I come across an equation ∂2u/∂x2 + ∂2u/∂y2 - k sinh(u) =0 How can I solve this equation analytically? Can anyone help me to solve this anlytically or using any mathematical packages like Mathematica or Maple.

 

I don't see that an analytical solution is possible. What evidence or indication do you have that suggests such a thing? Just to be clear:
u is a real valued function of x and y which belong to R2
and sinh(u) =(e^u - e^-u)/2. Is that correct?
Are there any boundry conditions?