1. Find all functions u_0(x)...

Find all functions $u_0(x)$ for which the problem

$u_x-2u=0$

$u(x,0)=u_0(x)$

$\displaystyle\int\frac{u_x}{u}dx=\int 2dx\Rightarrow \ln|u|=2x+g(y)$

$u(x,y)=\varphi(y)\exp{(2x)}$

$u(x,0)=\varphi(0)\exp{(2x)}=u_0(x)$

How is $\varphi(0)$ handled in this situation?

Thanks.

2. Assuming that $\varphi(0)$ is defined, it will just be a constant.