Show that

$$a(x, y)\left(\frac{d y}{d s}\right)^{2}-2 b(x, y) \frac{d x}{d s} \frac{d y}{d s}+c(x, y)\left(\frac{d x}{d s}\right)^{2}=0$$

along a characteristic curve $(x(s), y(s))$ of the $2^{\text {nd }}$-order pde

$$a(x, y) u_{x x}+2 b(x, y) u_{x y}+c(x, y) u_{y y}=f(x, y)$$