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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


This page titled 2.2: Quasilinear Equations is shared under a not declared license and was authored, remixed, and/or curated by Erich Miersemann.

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