Uniqueness of a solution to a 1st order PDE

Question

I'm asked to discuss the uniqueness of the solution for $$u_y+u_x=u-x-y$$ $$u(x,-2)=x$$ I've found the solution to be $$u(x,y)=x+y+2$$ But I don't know how to prove the uniqueness of the solution.

Answer 1

Take $$ v(x,y)=u(x,y)-x-y-2. $$ We shall show that $v\equiv 0$. The function $v$ satisfies $$ v_y+v_x=v, \quad v(x,-2)=0. $$ Fix $s\in\mathbb R$ and set $f(t)=v(s+t,t)$. Then $$ f'(t)=v_x+v_y=v=f(t), $$ and hence $$ f(t)=\mathrm{e}^{t+2} f(-2)=\mathrm{e}^{t+2} v(s-2,-2)=0. $$ Thus $v(s+t,s)=0$, for all $s,t\in\mathbb R$, and hence $v\equiv 0$.