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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.

Patrick Windance
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1 Answers1

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No, I do not think there is any other way since you lack the information needed to perform the usual trace-back to initial data. For a general point $(x, y)$, you have no idea what the corresponding characteristic equation is. The only thing you can do is follow the $x_5(y) = 5 + h(5) \cdot y = 5 + 80 \cdot y$ characteristic and observe that it goes through the point $(165, 2)$.

Dan Doe
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