
6.3. Equations Containing Power Functions and Arbitrary Parameters 597
5.
∂
2
w
∂t
2
= (ax + b)
∂
2
w
∂x
2
+ a
∂w
∂x
+ cw + Φ(x, t).
The substitution z = ax + b leads to an equation of the form 6.3.1.2:
∂
2
w
∂t
2
= a
2
∂
∂z
z
∂w
∂z
+ cw + Φ
z − b
a
, t
.
6.
∂
2
w
∂t
2
= (ax + b)
∂
2
w
∂x
2
+
1
2
a
∂w
∂x
+ cw + Φ(x, t).
The substitution z = 2
√
ax + b leads to the equation
∂
2
w
∂t
2
= a
2
∂
2
w
∂z
2
+ cw + Φ
z
2
− 4b
4a
, t
,
which is considered in Section 6.1.3.
7.
∂
2
w
∂t
2
= (a
2
x + b
2
)
∂
2
w
∂x
2
+ (a
1
x + b
1
)
∂w
∂x
+ (a
0
x + b
0
)w.
This is a special case of equation 6.5.3.4 with f (x) = a
2
x + b
2
, g(x) = a
1
x + b
1
, h(x) =
a
0
x + b
0
, and Φ ≡ 0.
Particular solutions:
w(x, t) = exp(kx)F
x + q
p
A sin
t
√
µ
+ B cos
t
√
µ
for µ > 0,
w(x, ...