Partial Differential Equations 0.15 alpha 

© Leon van Dommelen 

3.6 First order equations in more dimensions
The procedures of the previous subsections extend in a logical way to
more dimensions. If the independent variables are
, the first order quasilinear partial
differential equation takes the form

(3.6) 
where the and may depend on
and .
The characteristic equations can now be found from the ratios

(3.7) 
After solving different ordinary differential equations from this
set, the integration constant of one of them, call it can be
taken to be a general parameter function of the others,
and then substituting for
from the other
ordinary differential equation, an expression for results
involving one still undetermined, parameter function .
To find this remaining undetermined function, plug in whatever initial
condition is given, renotate the parameters of to
and express everything in terms of them to
find function
.