5.4. FIRST INTEGRALS AND LINEARIZATION 115
At this point, we need to find U such that the contact conditions (5.165) are satisfied, namely
U
p
D 0;
U
q
D 0;
U
u
D
p C1
q
q C 1
p
C ; (5.185)
U
x
D
q C 1
p
p;
U
y
D
p C1
q
q:
However, cross differentiation of (5.185) shows they are not consistent; thus, another choice
will be required. Next we try
X D x Cu; Y D y Cu; (5.186a)
P D A
q C 1
p
; Q D B
p C1
q
; (5.186b)
and determine whether the functions A and B can be chosen such that the contact conditions
can be satisfied. With (5.186), the contact conditions (5.165) become
U
p
D 0;
U
q
D 0;
U
u
D A
q C 1
p
B
p C1
q
C ; (5.187)
U
x
D A
q C 1
p
p;
U
y
D B
p C1
q
q;
or, after eliminating , the latter two of (5.187) become
U
x
C pU
u
D .p C 1/A
q C 1
p
C pB
p C1
q
; (5.188a)
U
y
C qU
u
D qA
q C 1
p
C .q C 1/B
p C1
q
: (5.188b)
Differentiating (5.188a) with respect to q (or (5.188b) with respect to p) gives
q
2
A
0
q C 1
p
p
2
B
0
p C1
q
D 0: (5.189)
If we let
p D
s C 1
rs 1
; q D
r C 1
rs 1
; (5.190)
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