So this PDE is a transport equation given as:
$u_x + xu_y = 0$ with initial condition that $u(x,0) = \exp(x)$.
My approach: Like usual I tried it to interpret this as a direction derivation, and found the curve $C_1$ (which guides me the direction to take for differenciation). Now since RHS is $0$, the value of $u$ of this curve should remain constant, but my calculation shows me the this curve intersect $x$-axis twice at different $x$ and so, the constant value should be $\exp(x_1)$ and $\exp(x_2)$ for $x_1 \neq x_2$. This appears to me as clear contradiction and hence I suspect the initial data. Could someone confirm my suspicion?
P.S If my explanation appears unclear, then try solving the above PDE by using characteristics method.