In addition to the second order equations of the type discussed in sections 2--3, systems of first order equations are also frequently encountered in computational science.

The current and voltage at position
**x** and time **t** in a
transmission line satisfy the first order equations

where **R**, **L**, **C** and **G** denote, respectively, resistance,
inductance, capacitance and leakage conductance per unit
length of transmission line.

(b) The first order system

governs the one dimensional flow of an ideal gas with velocity , density and pressure . is a physical constant determined by the specific heat of the gas.

Problems such as these present computational
scientists with systems of first order partial differential
equations. The general * quasilinear system* of * n* first order
partial differential equations in two independent variables
has the form

where , and may depend on
. If each and is
independent of , the system (15) is
called * almost linear*. If, in addition, each depends
linearly on ,

with and **S** functions of at most **x** and **t**, the system is
said to be * linear*. If for ,
the system
is called * homogeneous*. If **C, G, R** and **L** depend at most on **x**
and **t** the transmission line equations are linear. The ideal
gas equations are quasilinear.

In terms of the matrices and the column vectors , the system of equations (15) can be written as