
376 SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE
2
◦
. If the nonhomogeneous part of the equation can be represented in the form
Φ(x, t) =
N
X
n=1
g
n
(x)t
n
,
then there is a particular solution that is the sum of particular solutions of the form (1):
¯w(x, t) =
N
X
n=1
¯w
n
(x, t).
For example, if
Φ(x, t) = g(x)t + h(x),
where g(x) and h(x) are arbitrary functions, the original equation has a solution of the
form
¯w(x, t) = −tψ(x) −
Z
x
x
0
ψ(ξ) + h(ξ)
f(ξ)
(x − ξ) dξ,
ψ(x) =
Z
x
x
0
g(ξ)
f(ξ)
(x − ξ) dξ, x
0
is any.
For the structure of particular solutions for other Φ(x, t), see equation 3.8.6.5, Item 3
◦
.
By summing different solutions of the homogeneous equation (see ...