# Solution to a two dimensional transport PDE

I've trying to solve this PDE with a initial condition using the method of characteristics \begin{equation*} u_t + yu_x - xu_y = 0 , \quad u(0,x,y)=e^{-(x-1)^2-(y-1)^2} \end{equation*} but i'm struggling since the possible results that i have gotten don't satisfy the PDE.

I have tried solving $x'= y$ and $y'= -x$ but i only get something as mentioned above.

So, any idea on how to aproach to this problem?

The method of characteristics (below) leads to the solution : $$u(x,y,t)=e^{-x^2-y^2+2(x+y)\cos(t)+2(x-y)\sin(t)-2}$$
• could u please explain me why did u take $tan(c_2)$, i dont really understand why is that possible, thank you @JJacquelin – Alek Murt Mar 5 '17 at 16:21
• You can take any function that you want. $\Phi(X,Y)=F\left(g(X),h(Y)\right)$, any functions $\Phi,F,g,h$. A function of arbitrary function is an arbitrary function as well. – JJacquelin Mar 5 '17 at 18:05