
3.3. Equations Containing Power Functions and Arbitrary Parameters 331
6.
∂w
∂t
= ax
1−n
∂
2
w
∂x
2
.
This equation is encountered in diffusion boundary layer problems (see equation 3.9.1.3)
and is a special case of 3.8.6.1 with f(x) = ax
1−n
. In addition, it is a special case of
equation 3.3.5.1 with n = −1 and is an equation of the form 3.3.6.2 for n = −3 (in
both cases the equation is reduced to a constant coefficient equation). For n = 0, see
equation 3.3.4.1.
1
◦
. Particular solutions (A, B, and µ are arbitrary constants):
w(x) = Ax + B,
w(x, t) = Aan(n + 1)t + Ax
n+1
+ B,
w(x, t) = Aa(n + 1)(n + 2)tx + Ax
n+2
+ B,
w(x, t) = A
an(n + 1)t
2
+ 2tx
n+1
+
x
2n+2
a(n + 1)(2