5.6. REFERENCES 125
where P D P ./. If we introduce a stream function such that
D
x
; u D
t
; (5.266)
then (5.265a) is identically satisfied, whereas (5.265b) becomes
2
x
t t
2
t
x
tx
C
2
t
xx
2
x
P
0
.
x
/
xx
D 0: (5.267)
Determine the forms of P ./ such that (5.267) admits a first integral.
5.3. Show the PDE
2xu
y
u
xy
C u
2
y
u
yy
C x
2
u
xx
u
yy
u
2
xy
D u
2
y
(5.268)
admits the first integral
F
q
x
; u xp
1
2
q
2
; p
yq C q
2
x
D 0: (5.269)
Use this to find a contact transformation that linearizes the PDE.
5.4. Show the PDE
2xu
y
u
xy
C x
2
u
xx
u
yy
u
2
xy
D 1 Cu
2
y
(5.270)
admits the first integral
F
1 ˙ iq
x
; y ˙ i.xp u/
D 0: (5.271)
Use this to find a contact transformation that linearizes the PDE.
5.6 REFERENCES
[72] S. Lie, Neue integrations methode der Monge–Ampèreshen gleichung, Arch. Math. Kris-
tiania, 2, pp. 1–9, 1877. 110
[73] A. R. Forsyth, eory of Differential Equations, vol. 6, Cambridge University Press, Cam-
bridge, 1906. 110
[74] M. H. Martin, e propagation of a plane shock into a quiet atmosphere, Can. J. Math.,
5, pp. 37–39, 1953. DOI: 10.4153/CJM-1953-004-2. 107
[75] M. H. Martin, e Monge–Ampere partial differential equation rt s
2
C
2
D 0, Pac. J.
Math., 3, pp. 165–187, 1953. DOI: 10.2140/pjm.1953.3.165. 108
[76] S. V. Meleshko, Methods for Constructing Exact Solutions of Partial Differential Equations,
Springer, 2005. 109
[77] A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations,
Chapman and Hall/CRC, Boca Raton, FL, 2004. DOI: 10.1201/9780203489659. 109
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