
11.1. Third-Order Partial Differential Equations 951
11.1.4 O n e-Dimensional Equations Containing a Mixed Derivative
and the Second Derivative in t
1.
∂
2
w
∂t
2
− a
∂
2
w
∂x
2
− b
∂
3
w
∂t∂x
2
= 0.
This equation describes one-dimensional unsteady motions of viscous compressible baro-
tropic fluids.
1
◦
. Particular solutions:
w(x, t) = (A
1
t + A
2
)x
2
+ (B
1
t + B
2
)x +
1
3
aA
1
t
3
+ (aA
2
+ bA
1
)t
2
+ C
1
t + C
2
;
w(x, t) = e
βt
A
1
exp
−
βx
√
a + bβ
+ A
2
exp
βx
√
a + bβ
, a + bβ > 0;
w(x, t) = e
βt
A
1
cos
βx
p
|a + bβ|
+ A
2
sin
βx
p
|a + bβ|
, a + bβ < 0,
where A
n
, B
n
, C
n
, and β are arbitrary constants. The last solution is periodic in x.
2
◦
. Solutions periodic in t:
w(x, t) = e
−λx
A cos(ωt −µx) + B sin( ...