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# 2.2: Quasilinear Equations

Here we consider the equation

\label{quasi}
a_1(x,y,u)u_x+a_2(x,y,u)u_y=a_3(x,y,u).

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

• Integrated by Justin Marshall.