Characteristic equation of a PDE
Probably is a particular case of method of characteristics...
The following discussion is made for first order homogeneous linear PDEs, but according to this entry of wikipedia it can be generalized to any first order PDE (see section Fully nonlinear case).
Given a first order homogeneous linear PDE
$$ a_{1} \partial W / \partial x_{1}+\cdots+a_{n} \partial W / \partial x_{n}=0 \tag{1} $$where $a_k=a_k(x_1,\ldots,x_n)$, we look for solutions
$$ W: \mathbb{R}^n \rightarrow \mathbb{R}. $$We can consider a solution like "surfaces" $S$ given implicitly by $W=cte$. Their tangent planes are given by the kernel of $dW$.
If we define the vector field $A=a_1\partial x_1+a_2\partial x_2+\ldots+a_n \partial x_n$, a function $W$ is solution of (1) if $A(W)=0$, i.e., the integral curves of $A$ are contained in the surfaces $W=cte$.
The ODE
$$ \dot{x}=A(x) \tag{2} $$is called the characteristic system or characteristic equation, and their integral curves are called the characteristic curves of the original PDE. Solutions of the PDE are first integrals of the characteristic equation.
The ODE (2) is, for me, like a parametric expression of a line $r$ in linear algebra (it is the case when $A(x)$ is a constant vector field). On the contrary, equation (1) is similar to an implicit expression of all the planes containing the line $r$. The 1-form $dW$ is like an implicit equation
Method of characteristic
I think that method of characteristics is usually called to what follows.
The line $r$ above can be described by saying that it is made of vectors such that they are proportional to $(a_1,\ldots,a_n)$, that is,
$$ \frac{dx_1}{a_1}=\frac{dx_2}{a_2}=\cdots $$We can think of this expression as the symmetric form of a line in space ("expresión continua de la recta en el espacio" in Spanish).
From here, in the case of linear algebra, we recover several planes whose intersection gives the line $r$:
$$ a_2 x_1-a_1 x_2=0 $$ $$ a_3 x_2-a_2 x_3=0 $$ $$ ... $$In the differential context, we could recover an independent set of first integral if we were able to solve
$$ a_2 \mbox{d}x_1-a_1 \mbox{d}x_2=0 $$ $$ a_3 \mbox{d}x_2-a_2 \mbox{d}x_3=0 $$ $$ ... $$The solutions are independent first integrals of the ODE or functions that generate functionally the solutions of the PDE.
In the general case, i.e. $a_i$ depending on too many variables, the system cannot be solved, although is useful for numerical approximation of initial or boundary value problems.
Particular case
If in the PDE we look for solutions in the form of explicit functions
$$ W=x_n-f(x_1,\ldots,x_{n-1}) $$we get
$$ a_{1} \partial W / \partial x_{1}+\cdots+a_{n-1} \partial W / \partial x_{n-1}=a_n $$which is the case for inhomogeneous linear PDEs.
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Author of the notes: Antonio J. Pan-Collantes
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