
1128 METHODS FOR FIRST-ORDER LINEAR PDES
Equation (13.1.2.5) can be treated as an ordinary differential equation for w = w(x)
with parameter u. The general solution of Eq. (13.1.2.5) has the form
w = E
Z
¯
h
0
(x, u)
¯
f(x, u)
dx
E
+ Φ(u)
, E = exp
Z
¯
h
1
(x, u)
¯
f(x, u)
dx
,
where Φ is an arbitrary function; in the integration, u is considered a parameter. To find a
general integral of Eq. (13.1.2.1), one should compute the integrals in the last relation and
then return to the original variables x and y.
◮ Classical and generalized C auchy problems. Solution methods.
1
◦
. The classical and generalized Cauchy problems for Eq. (13.1.2.1) are stated in the same
manner ...