I am trying to Solve a Cauchy Problem Given by
$$\frac{\partial{u}}{\partial t}+u\frac{\partial{u}}{\partial x}=0\;\;with \;\;u(x,0)=u_0(x)\;\;where \;\;x\in \mathbb{R}$$
The Auxiliary equation is $$\frac{dx}{u}=\frac{dt}{1}=\frac{du}{0}$$
so from this one Characteristic equation is given by $u=c_1$. But i can't figure out the second one
Please help
Thankyou.