
6.4. Equations Containing the First Time Derivative 607
where
α =
m − 2b
2(m + 2)
, β =
m + 2b
2(m + 2)
, Γ(z) =
Z
∞
0
e
−s
s
z−1
ds.
2
◦
. Solution for b =
1
2
m:
w(x, t)= f
x+
2
m+2
t
m+2
2
+
2t
m+2
Z
1
0
g
x+
2
m+2
t
m+2
2
(2ξ−1)
(1−ξ)
−
m
m+2
dξ.
3
◦
. Solution for b = −
1
2
m:
w(x, t)= f
x−
2
m+2
t
m+2
2
+
2t
m+2
Z
1
0
g
x+
2
m+2
t
m+2
2
(2ξ−1)
(1−ξ)
−
m
m+2
dξ.
⊙ Literature: M. M. Smirnov (1975).
13. (b + x)
2
∂
2
w
∂t
2
= a
2
∂
∂x
hh
(b + x)
2
∂w
∂x
ii
.
General solution:
w(x, t) =
f(x + at) + g(x − at)
b + x
,
where f(y) and g(z) are arbitrary functions.
6.4 Equations Containing the First Time Derivative
6.4.1 Equations of the Form
∂
2
w
∂t
2
+k
∂w
∂t
= a
2
∂
2
w
∂x
2
+b
∂w
∂x
+cw+Φ(x, t)
1.
∂
2
w
∂t
2
+ k
∂w
∂t
= a
2
∂
2
w
∂x
2
+ Φ(x, t).
For Φ(x, t) ≡ 0, this equation ...