# Clarification on equations and terminology of characteristic curves

I am currently studying the textbook Partial Differential Equations – An introduction, second edition, by Walter A. Strauss. The section The Variable Coefficient Equation of chapter 1 says the following:

The equation $$u_x + y u_y = 0 \label{4}\tag{4}$$ is linear and homogeneous but has a variable coefficient ($$y$$). We shall illustrate for equation \eqref{4} how to use the geometric method somewhat like Example 1. The PDE \eqref{4} itself asserts that the directional derivative in the direction of the vector $$(1, y)$$ is zero. The curves in the $$xy$$ plane with $$(1, y)$$ as tangent vectors have slopes $$y$$ (see Figure 3). Their equations are $$\dfrac{dy}{dx} = \dfrac{y}{1} \label{5}\tag{5}$$ This ODE has the solutions $$y = Ce^x \label{6}\tag{6}$$ These curves are called the characteristic curves of the PDE \eqref{4} . As $$C$$ is changed, the curves fill out the $$xy$$ plane perfectly without intersecting. On each of the curves $$u(x, y)$$ is a constant because $$\dfrac{d}{dx}u(x, Ce^x) = \dfrac{\partial{u}}{\partial{x}} + Ce^x \dfrac{\partial{u}}{\partial{y}} = u_x + yu_y = 0.$$

Example 3. then says the following:

Solve the PDE
$$u_x + 2xy^2 u_y = 0. \tag{8}$$ The characteristic curves satisfy the ODE $$dy/dx = 2xy^2/1 = 2xy^2$$. To solve the ODE, we separate variables: $$dy/y^2 = 2x \ dx$$; hence $$-1/y = x^2 - C$$, so that $$y = (C - x^2)^{-1}$$
These curves are characteristics. Again $$u(x, y)$$ is a constant on each curve.

This seems like it's poorly written, so I'm a bit confused with the terminology here. At the beginning, the author seems to say that the curves (characteristic curves?) have equations $$\dfrac{dy}{dx} = \dfrac{y}{1}$$ (not $$y = Ce^x$$?). Then, for example 3, the author says that the characteristics (characteristic curves?) have equations $$y = (C - x^2)^{-1}$$; but it seems to me that this corresponds with $$y = Ce^x$$, not $$\dfrac{dy}{dx} = \dfrac{y}{1}$$. So what's going on here?

A curve is just a general terminology. Usually there is a simple explicit parametric representation $$(x(t),y(t))$$ or as given in the examples here an explicit representation in the form $$(x,y(x))$$.

Note, that the curves are not given in either forms at first but rather described via ODE equations. Those needed to be solved at first to get explicit representations.

Furthermore, a characteristic curve has another property: the solution $$u=u(x,y)$$ is constant on that specific curve.

Just to be sure. I think what confused you might be the part "Their equations are $$\frac{dy}{dx}=\frac{y}{1}$$ ...". That is just saying that the curves are described by solutions to that specific ODE.

• So "characteristics" and "characteristic curves" are the same thing? Nov 1, 2020 at 22:48
• @ThePointer Yes.
– jack
Nov 2, 2020 at 13:13
• So what is the connection between $y = Ce^x$ and $y = (C - x^2)^{-1}$, because doesn't the author suggest that these are both characteristic curves? Nov 2, 2020 at 13:15
• but to different PDEs
– jack
Nov 2, 2020 at 14:28