
3.8. Equations Containing Arbitrary Functions 361
3.
∂w
∂t
= a
∂
2
w
∂x
2
+ xf(t)w.
1
◦
. Particular solutions (A and λ are arbitrary constants):
w(x, t) = A exp
h
xF (t) + a
Z
F
2
(t) dt
i
, F (t) =
Z
f(t) dt,
w(x, t) = A
h
x + 2a
Z
F (t) dt
i
exp
h
xF (t) + a
Z
F
2
(t) dt
i
,
w(x, t) = A exp
h
xF (t) + λx + aλ
2
t + a
Z
F
2
(t) dt + 2aλ
Z
F (t) dt
i
.
2
◦
. The transformation
w(x, t) = u(z, t) exp
h
xF (t) + a
Z
F
2
(t) dt
i
, z = x + 2a
Z
F (t) dt,
where F (t) =
Z
f(t) dt, leads to a constant coefficient equation, ∂
t
u = a∂
zz
u, which is
considered in Section 3.1.1.
4.
∂w
∂t
= a
∂
2
w
∂x
2
+ x
2
f(t)w.
This is a special case of equation 3.8.7.5.
5.
∂w
∂t
= a
∂
2
w
∂x
2
+
f(x) + g(t)
w.
1
◦
. There are particular solutions in the product ...