Details

Multyplying the latter equation by
inline
y
z
]]>
and integrating with respect to
inline
z
]]>
one can obtain
y
z
2
= −
y
4
+
c
0
y
2
−
c
1
y +
c
2
.
]]>
Let
inline
α , β , γ
]]>
and
inline
δ
]]>
inline
( δ ≤ γ ≤ β ≤ α )
]]>
be the solutions of the 4^{th} order algebraic equation
y
4
−
c
0
y
2
+
c
1
y −
c
2
= 0.
]]>

Details

One can obtain then
∫
0
q
d x
(
1 −
x
2
) (
1 −
m
2
x
2
)
=
1
2
(
β − δ
) (
α − γ
)
(
z −
z
0
) .
]]>
There is an elliptic integral of the first kind in the left hand side of the latter equation.

Details

Let us recall the following well-known relations for hyperbolic functions
c
h
2
(
k z
) − s
h
2
(
k z
) = 1,
]]>
c
h
2
(
k z
) + s
h
2
(
k z
) = c h (
2 k z
) ,
]]>

- large-amplitude internal waves in the ocean

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- Modified Korteweg-de Vries equation in EqWorld, the world of mathematical equations