
3.8. Equations Containing Arbitrary Functions 381
3
◦
. The structure of particular solutions ¯w(x, t) of the nonhomogeneous equation 3.8.6.5
for some functions Φ(x, t) is presented in Table 3.4.
Remark 3.7. The homogene ous equation (with Φ ≡ 0) admits all the particular solutions spec-
ified in Table 3.4. In this case, n should be assumed a n integer an d β and λ arbitrary nu mbers.
TABLE 3.4
Structure of particular solutions of linear nonhomogeneous equations of special form
No Functions Φ(x, t) Form of particular solutions ¯w(x, t) Remarks
1
ϕ(x)t
n
n
P
m=0
ψ
m
(x)t
m
n is an integer; the equations
for ψ
m
(x) are solved consecutively,
starting with m = n
2
ϕ(x)