# Linear and Quasi-Linear PDEs

First-order PDEs are usually classified as linear, quasi-linear, or nonlinear. The first two types are discussed in this tutorial.

A first-order PDE for an unknown function 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 quasi-linear is said to be *nonlinear*.

For convenience, the symbols , , and are used throughout this tutorial 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 , , and . Since there is no term free of , , or , the PDE is also homogeneous.

The *transport equation* is a good example of a linear first-order homogeneous PDE with constant coefficients.

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 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.

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Now consider some examples of *first-order quasi-linear PDEs*.

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*Burgers’ equation*is an important example of a quasi-linear PDE.

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The term makes this equation quasi-linear.

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A practical consequence of quasi-linearity is the appearance of shocks and steepening and breaking of solutions. Thus, although the procedures for finding general solutions to linear and quasi-linear PDEs are quite similar, there are sharp differences in the nature of the solutions.