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Linear and Quasilinear PDEs

First-order PDEs are usually classified as linear, quasilinear, or nonlinear. The first two types will be discussed in this section.

Here is a linear homogeneous first-order PDE with constant coefficients. For convenience, the symbols z, p, and q are used throughout this section to denote the unknown function and its partial derivatives.

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The equation is linear because the left-hand 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  (2y-3x).

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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 first-order homogeneous PDE with constant coefficients.

In this transport equation,  for convenience.

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Note that the solution to the transport equation is constant on any straight line of the form  in the plane. These straight lines are called the base characteristic curves. The equation  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  ,  , and  , 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.

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The first part of the solution,  , 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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This verifies the solution.

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Here is a linear inhomogeneous PDE with variable coefficients.

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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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The term quasilinear means that the PDE is linear in the partial derivatives p and q, but may have nonlinear combinations of z, p, and q. Some examples of first-order quasilinear PDEs follow.

This PDE is quasilinear because of the term z^2 on the right-hand side.

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This verifies the solution.

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Burgers' equation is an important example of a quasilinear PDE. The term  makes this equation quasilinear.

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This solves the equation. The warning message is displayed because the solution is only available in implicit form, because the characteristic ODEs cannot be decoupled.

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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.