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
