Consider the first order quasi-linear equation with initail condition for a function $u(x,y)$ of two variables $x, y$ :
$$\left\{ \begin{align} & {{u}_{y}}+u{{u}_{x}}=0 \\ & u\left( x,0 \right)=h\left( x \right)\text{ },\text{ }x\in R \\ \end{align} \right.$$
where $h:R\to R$ is a smooth increasing function with $h(5)=80$, ${h}'(5)=3$. Find the values of ${{u}_{x}}\left( 165,2 \right)$ and ${{u}_{y}}\left( 165,2 \right)$.
My attempt:
By method of characteristics, we get:
$$u\left( x,y \right)=h\left( x-uy \right)$$
and also
$${{u}_{x}}=\frac{{h}'\left( x-uy \right)}{1+{h}'\left( x-uy \right)y}$$
$${{u}_{y}}=\frac{-u{h}'\left( x-uy \right)}{1+{h}'\left( x-uy \right)y} .$$
By observation, we can get
$$80=u(165,2)=h(165-u(165,2)2)=h(5)=80 $$
and solve ${u}_{x},{u}_{y}$ accordingly.
Is there a way to derive $u(165,2)=80$ instead of "by observation"? Please help.
