I have been trying to solve the following problem:
Let $u(x,y)$ be the solution to the Cauchy Problem $xu_{x}+u_{y}=1,\;\;u(x,0)=2\ln(x),\quad x>1.$ Then $u(e,1)=?$ I was trying to solve it by Lagrange's method but could not progress. Can someone point me in the right direction? (A certain property or theorem that I have to use to find out $u(x,y)$.)