# How to correctly DSolve it?

Solving a PDE with Mathematica, I obtain

sol = DSolve[Sqrt[D[u[x, y], x]] + Sqrt[D[u[x, y], y]] == x, u, {x, y}]


{{u -> Function[{x, y}, x^3/3 + 2 x^2 y + (x^2 C[1])/2 + x y C[1] + (x C[1]^2)/4 + (y C[1]^2)/4 - x y Sqrt[(2 x + C[1])^2] + C[2]]}}

with a warning (not an error) "DSolve::nlpde: Solution requested to nonlinear partial differential equation. Trying to build a special solution". Unfortunately, the result is not correct, as

FullSimplify[(Sqrt[D[u[x, y], x]] + Sqrt[D[u[x, y], y]] /.
sol) /. {C[1] -> 1}, Assumptions -> x > 0]


(1 + x)

shows. The question arises: how to correctly DSolve this PDE?

• Mathematica v12.2 doesn't evaluate sol ! What's your version ? Jun 8 at 13:08
• 12.3 on Windows 10 Pro. You may try it in MathematicaOnline. Jun 8 at 13:14
• Mathematica v12.2 gives the same message but doesn't find a special solution. Perhaps more trustworthy than the wrong result of v12.3? Jun 8 at 13:20
• BTW, Maple correctly finds a special solution $$u\! \left(x,y\right)=\frac{x^{3}}{3}-\sqrt{\textit{_}c_{2}}\, x^{2}+\textit{_}c_{2} x+\textit{_}\mathit{C1}+\textit{_}c_{2} y+\textit{_}\mathit{C2}.$$ Jun 8 at 13:28