Partial differential equation

I am trying to solve following PDE using DSolve but there is no result upon completion-

DSolve[D[-A[r,t],t]D[A[r,t],r]+A[r,t]D[D[A[r,t],r],t]==0,A[r, t], {r, t}]

• DSolve[]'s support for PDE's is still somewhat limited, so don't be surprised if some things don't work yet :P. Try alternative -> Maple. – Mariusz Iwaniuk Feb 9 '18 at 12:59
• The approach A[r,t]=R[r]T[t](separation of variables) solves the pde without any restriction concerning R[r],T[t] ??? – Ulrich Neumann Feb 9 '18 at 13:04

Your pde leads to $1=1$. So any two random functions $R(r)$ and $T(t)$ give a solution.

Solve $$-u_{t}u_{r}+u\ u_{rt}=0$$

Let $u=R\left( r\right) T\left( t\right)$ then the above becomes

\begin{align*} -\left( RT^{\prime}\right) \left( R^{\prime}T\right) +\left( RT\right) \ \left( R^{\prime}T^{\prime}\right) & =0\\ R^{\prime}T^{\prime} & =R^{\prime}T^{\prime}\\ 1 & =1 \end{align*}

So try any random functions, $R(r)$ and $T(t)$, they will be a solution

ClearAll[r,t,solvePDE]
solvePDE[R0_,T0_,r_,t_]:=Module[{u},
u=R0 T0;
Simplify[D[-u,t]D[u,r]+u D[D[u,r],t]]
]

solvePDE[Sin[r],Exp[-t],r,t]
(* 0 *)

solvePDE[Sin[r]*Cos[r]+2*r,Exp[-t]+t+Tan[t],r,t]
(* 0 *)

solvePDE[r^3, t^8, r, t]
(* 0 *)


May be Mathematica should just have returned $F1(r) F2(t)$ as an answer?