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Mathematics LibreTexts

2.2: Quasilinear Equations

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    2134

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    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.

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