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d
x
d
u
2
x
2
)(1
k
2
x
2
);
=(1
−
−
(C.9)
y
=cn
u,
cn
u
=
dcn
u
d
u
=
−
sn
u
dn
u,
(C.10)
d
y
d
u
2
=(1
− y
2
)(1
− k
2
+
k
2
y
2
);
(C.11)
z
=dn
u,
dn
u
=
ddn
u
d
u
k
2
sn
u
cn
u,
=
−
(C.12)
d
z
d
u
2
z
2
)[
z
2
k
2
])
.
=(1
−
−
(1
−
(C.13)
End of Lemma.
Lemma C.3 (Differential equations of elliptic functions).
The elliptic functions sn (
u, k
), cn (
u, k
), and dn (
u, k
) satisfy the algebraic differential equations
of second order that follow:
x
=sn
u,
(C.14)
x
=
d
2
x
(1 +
k
2
)
x
+2
k
2
x
3
;
d
u
2
=
−
(C.15)
y
=cn
u,
(C.16)
y
=
d
2
y
d
u
2
=
−
(1
−
2
k
2
)
y
+2
k
2
y
3
;
(C.17)
z
=dn
u,
(C.18)
z
=
d
2
z
k
2
)
z
2
z
3
.
d
u
2
=+(2
−
−
(C.19)
End of Lemma.
Proof (Proof of formulae (
C.8
)and(
C.9
)).
x
:= sn
u,
√
1
x
2
=cn
u,
√
1
−
−
k
2
x
2
=dn
u,
(C.20)
d
u/
d
x
=
(1
d
x
d
u
=
1
−
x
2
)(1
−
k
2
x
2
)
,
d
x
d
u
=cn
u
dn
u
q
.
e
.
d
.
(C.21)
d
x
d
u
2
x
2
)(1
k
2
x
2
)q
.
e
.
d
.
=(1
−
−
End of Proof (Proof of formulae (
C.8
)and(
C.9
)).
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