
316 SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE
1
◦
. Particular solutions (A, B, and µ are arbitrary constants):
w(x, t) = (Ax + B) exp
1
2
bt
2
+ ct
,
w(x, t) = A(x
2
+ 2at) exp
1
2
bt
2
+ ct
,
w(x, t) = A exp
µx +
1
2
bt
2
+ (c + aµ
2
)t
,
w(x, t) = A exp
1
2
bt
2
+ (c − aµ
2
)t
cos(µx),
w(x, t) = A exp
1
2
bt
2
+ (c − aµ
2
)t
sin(µx).
2
◦
. The substitution w(x, t) = u(x, t) exp
1
2
bt
2
+ct
leads to a constant coefficient equa-
tion, ∂
t
u = a∂
xx
u, which is considered in Section 3.1.1.
9.
∂w
∂t
= a
∂
2
w
∂x
2
+ bt
k
w.
This is a special case of equation 3.8.1.1 with f (t) = bt
k
.
1
◦
. Particular solutions (A, B, and µ are arbitrary constants):
w(x, t) = (Ax + B) exp
b
k + 1
t
k+1
,
w(x, t)