
386 SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE
2.
∂w
∂t
= x
2
f(t)
∂
2
w
∂x
2
+ xg(t)
∂w
∂x
+ h(t)w.
The substitution x = ±e
ξ
leads to an equation of the form 3.8.7.3:
∂w
∂t
= f (t)
∂
2
w
∂ξ
2
+
g(t) − f (t)
∂w
∂ξ
+ h(t)w.
3.
∂w
∂t
= x
2
n(t)
∂
2
w
∂x
2
+x
f(t) ln x+g(t)
∂w
∂x
+
h(t) ln
2
x+s(t) ln x+p(t)
w.
The substitution z = ln x leads to an equation of the form 3.8.7.5:
∂w
∂t
= n(t)
∂
2
w
∂z
2
+
zf (t) + g(t) − n(t)
∂w
∂z
+
z
2
h(t) + zs(t) + p(t)
w.
4.
∂w
∂t
= x
4
f(t)
∂
2
w
∂x
2
+ g(t)w.
1
◦
. Particular solutions (A, B, and λ are arbitrary constants):
w(x, t) = (Ax + B) exp
Z
g(t) dt
,
w(x, t) =
2Ax
Z
f(t) dt + Bx +
A
x
exp
Z
g(t) dt
,
w(x, t) = Ax exp
λ
2
Z
f(t) dt +
Z
g(t) dt +
λ
x
.
2
◦
. The transformation ...