
1148 SECOND-ORDER LINEAR PDES: CLASSIFICATION, PROBLEMS, PARTICULAR SOLUTIONS
TABLE 14.4
Homogeneous linear partial differential equations that admit multiplicative separable solutions
No.
Form of Eq. (14.3.1.1) Form of particular solutions
1
Equation coefficients are
constant
w(x, t) = A exp(λt + β
1
x
1
+ ··· + β
n
x
n
),
λ, β
1
, . . . , β
n
are related by an algebraic equation
2
Equation coefficients are
independent of time t
w(x, t) = e
λt
ψ(x),
λ is an arbitrary constant, x = {x
1
, . . . , x
n
}
3
Equation coefficients are independent
of the coordinates x
1
, . . . , x
n
w(x, t) = exp(β
1
x
1
+ ··· + β
n
x
n
)ψ(t),
β
1
, . . . , β
n
are arbitrary constants
4
Equation coefficients are independent ...