# PDE with variable coefficient equation

Solving $U_x+yU_y=0$.The curves in the x,y plane with (1,y) as tangent vectors have slopes y.
Their equations are $dy/dx=y$.This Ode has the solution $y=Ce^x$.

Hence $u(x,y)=u(x,Ce^x)=U_x+Ce^xU_y=0$.
After this the book says $u(x,Ce^x)=u(0,Ce^0)=u(0,C)$ is independent of x.Why is $U(0,C )$ selected?Why specifically 0? and what's the need to use it

Perhaps since $y=Ce^x$, then $u$ is independent of $y$, so $u_y=0$, and the pde becomes $u_x=0$, and hence, $u$ is constant in $x$.

• :I don't understand this.Can you please explain why x=0 is plugged in for $u(x,Ce^x)$?Why x=0 is selected? – clarkson Jun 14 '14 at 12:58