
3.8. Equations Containing Arbitrary Functions 371
4.
∂w
∂t
= ax
2
∂
2
w
∂x
2
+ ln
2
x f(t)w.
This is a special case of equation 3.8.8.3.
5.
∂w
∂t
= ax
n
∂
2
w
∂x
2
+ xf(t)
∂w
∂x
.
1
◦
. Particular solutions (A and B are arbitrary constants):
w(x, t) = Ax exp
Z
f(t) dt
+ B,
w(x, t) = Ax
2−n
F (t) + Aa(n −1)(n −2)
Z
F (t) dt,
where
F (t) = exp
(2 − n)
Z
f(t) dt
.
2
◦
. On passing from t, x to the new variables
z = xF (t), τ = a
Z
F
2−n
(t) dt,
where
F (t) = exp
Z
f(t) dt
,
we obtain an equation of the form 3.3.6.6:
∂w
∂τ
= z
n
∂
2
w
∂z
2
.
6.
∂w
∂t
= ax
n
∂
2
w
∂x
2
+ xf(t)
∂w
∂x
+ bw.
The substitution w(x, t) = e
bt
u(x, t) leads to an equation of the form 3.8.4.5:
∂u
∂t
= ax
n
∂
2
u
∂x
2
+ xf(t)
∂u
∂x
.
7.
∂w
∂t
= ax
2n
∂
2
w
∂x
2
+
√
a x
n
√
a