
9.4. Other Equations 851
where Φ(x) and Ψ(y) are determined by the following second-order linear ordinary differ-
ential equations (C is an arbitrary constant):
(fΦ
′
x
)
′
x
− βΦ = C, f = f(x),
(gΨ
′
y
)
′
y
− βΨ = −C, g = g(y).
In the special case β = 0, the solutions of these equations can be represented as
Φ(x) = C
Z
x dx
f(x)
+ A
1
Z
dx
f(x)
+ B
1
,
Ψ(y) = −C
Z
y dy
g(y)
+ A
2
Z
dy
g(y)
+ B
2
,
where A
1
, A
2
, B
1
, and B
2
are arbitrary constants.
9.4.4 Other Equations Arising in App lications
1. y
∂
2
w
∂x
2
+
∂
2
w
∂y
2
= 0.
Tricomi equation. It is used to describe near-sonic flows of gas.
1
◦
. Particular solutions:
w = Axy + Bx + Cy + D,
w = A(3x
2
− y
3
) + B(x
3
− xy
3
) + C(6yx
2
− y
4
),
where A, B, C, and