
3.4. Equations Containing Exponential Functions and Arbitrary Parameters 341
7.
∂w
∂t
= a
∂
2
w
∂x
2
+ (be
βt
+ ce
λt
)w.
This is a special case of equation 3.8.1.1 with f (t) = be
βt
+ ce
λt
.
1
◦
. Particular solutions (A, B, and ν are arbitrary constants):
w(x, t) = (Ax + B) exp
b
β
e
βt
+
c
λ
e
λt
,
w(x, t) = A(x
2
+ 2at) exp
b
β
e
βt
+
c
λ
e
λt
,
w(x, t) = A exp
νx + aν
2
t +
b
β
e
βt
+
c
λ
e
λt
.
2
◦
. The substitution w(x, t) = u(x, t) exp
b
β
e
βt
+
c
λ
e
λt
leads to a constant coefficient
equation, ∂
t
u = a∂
xx
u, which is considered in Section 3.1.1.
8.
∂w
∂t
= a
∂
2
w
∂x
2
+ (be
βx
+ ce
λt
+ d)w.
The substitution w(x, t) = u(x, t) exp
c
λ
e
λt
leads to an equation of the form 3.4.1.3:
∂u
∂t
= a
∂
2
u
∂x
2
+ (be
βx
+ d)u.
9.
∂w
∂