Skip to main content
Mathematics LibreTexts

2.0: Prelude to First Order Equations

  • Page ID
    3445
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    For a given sufficiently regular function \(F\) the general equation of first order for the unknown function \()\) is

    $$F(x,u,\nabla u)=0\]

    in \(n\). The main tool for studying related problems is the theory of ordinary differential equations. This is quite different for systems of partial differential of first order. The general linear partial differential equation of first order can be written as

    $$\sum_{i=1}^na_i(x)u_{x_i}+c(x)u=f(x)\]

    for given functions \(a_i,\ c\) and \(f\). The general quasilinear partial differential equation of first order is

    $$\sum_{i=1}^na_i(x,u)u_{x_i}+c(x,u)=0.\]

    Contributors and Attributions


    This page titled 2.0: Prelude to First Order Equations is shared under a not declared license and was authored, remixed, and/or curated by Erich Miersemann.

    • Was this article helpful?