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Differential Equations by Pearson
book

Differential Equations by Pearson

by E. Rukmangadachari
May 2024
Intermediate to advanced content levelIntermediate to advanced
472 pages
13h 25m
English
Pearson India
Content preview from Differential Equations by Pearson
Partial Diff erential Equations 8-101
The complete integral is
222 2
().zay axb=++
Example 8.13.25 Find the complete integral of
1
2
(1 ) .px qy z pq+= +
Solution Charpit’s auxiliary equations give
11
22
or
(1 ) (1 )
dp dq dp dq
pq
p p pq q q pq
==
−+ −+
paq=
Substituting in the given equation we get
1
2
2
()(1)qax y z aq+= +
or
[]
2222
()q ax y az z+− =
[]
1
22
2
()
z
q
ax y az
⇒=
+−
Hence we get
()dz p dx q dy a dx dy q=+= +
[]
11
22
22
22
()
()
adx dy
dz du
z
uz
ax y az
+
∴= =
+−
where
au ax y=+
or
1
22
2
()du u z
dz z
=
Putting
uvz=
we have
2
1
dv
zvv
dz
=−−
2
2
1
1
dz dv
vvdv
z
vv
⎡⎤
==−−+
⎣⎦
−−
Hence the complete integral is
2
22
1
log 1 log ( 1)
22 2
vv
zv vvc++ −+ + −=
where
().vuz axyza==+
Ch08.indd ...
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Publisher Resources

ISBN: 9781299487406Publisher Website