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Mathematics LibreTexts

2.2: Quasilinear Equations

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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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