Linear and QuasiLinear PDEs
Firstorder PDEs are usually classified as linear, quasilinear, or nonlinear. The first two types are discussed in this tutorial.
A firstorder PDE for an unknown function
u (x,y) is said to be
linear if it can be expressed in the form
The PDE is said to be
quasilinear if it can be expressed in the form
A PDE which is neither linear nor quasilinear is said to
nonlinear.
For convenience, the symbols
z,
p, and
q are used throughout this tutorial to denote the unknown function and its partial derivatives.
Here is a linear homogeneous firstorder PDE with constant coefficients. 
The equation is linear because the lefthand side is a linear polynomial in
z,
p, and
q. Since there is no term free of
z,
p, or
q, the PDE is also homogeneous.
As mentioned earlier, the general solution contains an arbitrary function C[1] of the argument .
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This verifies that the solution is correct.
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Particular solutions of the homogeneous PDE are obtained by specifying the function C[1].
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Here is a plot of the surface for this particular solution.
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The
transport equation is a good example of a linear firstorder homogeneous PDE with constant coefficients.
In this transport equation, c=1 for convenience.
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Note that the solution to the transport equation is constant on any straight line of the form
yx+ in the plane. These straight lines are called the
base characteristic curves. The equation
yx+ defines a plane in three dimensions. The intersections of these planes with the solution surface are called
characteristic curves. Since the characteristic curves are solutions to a system of ODEs, the problem of solving the PDE is reduced to that of solving a system of ODEs for
x (t),
y (t), and
u (t), where
t is a parameter along the characteristic curves. These ODEs are called characteristic ODEs.
The solution to an inhomogeneous PDE has two components: the general solution to the homogeneous PDE and a particular solution to the inhomogeneous PDE.
This is a linear inhomogeneous PDE of the first order. 
The first part of the solution, 10+x+y, is the particular solution to the inhomogeneous PDE. The rest of the solution is the general solution to the homogenous equation.
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Here is a linear homogeneous PDE with variable coefficients.
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Here is a linear inhomogeneous PDE with variable coefficients. 
The solution is once again composed of the general solution to the homogeneous PDE and a particular solution, Sin[x], to the inhomogeneous PDE.
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Now consider some examples of
firstorder quasilinear PDEs.
This PDE is quasilinear because of the term z^{2} on the righthand side.
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It can be written using the notation introduced earlier. 
The term
z q makes this equation quasilinear.
This verifies the solution to Burgers' equation.
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A practical consequence of quasilinearity is the appearance of shocks and steepening and breaking of solutions. Thus, although the procedures for finding general solutions to linear and quasilinear PDEs are quite similar, there are sharp differences in the nature of the solutions.