# Geometric View of First-Order Quasilinear PDEs

Theorem 1 in page 4 of the book Numerical Solution of Partial Differential Equations in Science and Engineering by L. Lapidus:

The general solution of the quasilinear PDE $$a(x,y,u)u_x + b(x, y,u)u_y = c(x, y,u),$$ is given by $$G(v,w)=0,$$ where $G$ is an arbitrary function and where $v(x, y, u) = c_1$, and $w(x,y,u)=c_2$ form a solution of the equations $$\frac{dx}{a}=\frac{dy}{b}=\frac{du}{c}.$$

My question is how can I prove this theorem?

In that book, it explains how can interpret quasilinear PDEs geometrically, but I can't understand how to obtain $\frac{dx}{a}=\frac{dy}{b}=\frac{du}{c}$.

Assume that $u(x,y)$ is a solution then $$du=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy$$ $$\Rightarrow \frac{\partial u}{\partial y}=\frac{du-\frac{\partial u}{\partial x}dx}{dy}$$ The equation becomes $$a\frac{\partial u}{\partial x}+b\frac{du-\frac{\partial u}{\partial x}dx}{dy}=c$$ $$a\frac{\partial u}{\partial x}dy+bdu-b\frac{\partial u}{\partial x}dx=cdy$$ $$\frac{\partial u}{\partial x}\bigg(ady-bdx\bigg)=cdy-bdu$$ The equation holds for $$(1) ady-bdx=0\Rightarrow \frac{dy}{b}=\frac{dx}{a}$$ $$(2) cdy-bdu=0\Rightarrow \frac{du}{c}=\frac{dy}{b}$$

• Thanks @Occupy for your question, can i ask one another question? How G is answer of the pdes? i mean a construction proof. May 21, 2016 at 2:12
• @user626 You can check the Cauchy problem to get more insight. May 21, 2016 at 10:49
• @ccupy-gezi I read Cauchy problem but i can't find relation between my question and it. May 28, 2016 at 9:01