2.2: Quasilinear Equations
( \newcommand{\kernel}{\mathrm{null}\,}\)
Here we consider the equation
\begin{equation} \label{quasi} a_1(x,y,u)u_x+a_2(x,y,u)u_y=a_3(x,y,u). \end{equation}
The inhomogeneous linear equation
$$a_1(x,y)u_x+a_2(x,y)u_y=a_3(x,y)\]
is a special case of (\ref{quasi}).
One arrives at characteristic equations x'=a_1,\ y'=a_2,\ z'=a_3 from (\ref{quasi}) by the same arguments as in the case of homogeneous linear equations in two variables. The additional equation 3 follows from
\begin{eqnarray*} z'(\tau)&=&p(\lambda)x'(\tau)+q(\lambda)y'(\tau)\\ &=&pa_1+qa_2\\ &=&a_3, \end{eqnarray*}
see also Section 2.3, where the general case of nonlinear equations in two variables is considered.
Contributors and Attributions
Integrated by Justin Marshall.