
346 SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE
4.
∂w
∂t
= ax
2
∂
2
w
∂x
2
+
be
βt
+ c
w.
This is a special case of equation 3.8.8.2 with f (t) = a, g(t) = 0, and h(t) = be
βt
+ c.
5.
∂w
∂t
= ax
4
∂
2
w
∂x
2
+
be
βt
+ c
w.
This is a special case of equation 3.8.8.4 with f (t) = a and g(t) = be
βt
+ c.
6.
∂w
∂t
= ax
n
∂
2
w
∂x
2
+ x
be
βt
+ c
∂w
∂x
.
This is a special case of equation 3.8.4.5 with f (t) = be
βt
+ c.
7.
∂w
∂t
= ax
n
∂
2
w
∂x
2
+ bxe
βt
∂w
∂x
+ ce
µt
w.
This is a special case of equation 3.8.8.7 with f (t) = a, g(t) = be
βt
, and h(t) = ce
µt
.
3.4.5 Equations of the Form
∂w
∂t
= ae
βx
∂
2
w
∂x
2
+ f (x, t)
∂w
∂x
+ g(x, t)w
1.
∂w
∂t
= ae
βx
∂
2
w
∂x
2
.
This is a special case of equation 3.8.6.1 with f