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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
5-18 Diff erential Equations
5.2 GENERAL SOLUTION OF
2
2
dy dy
PQyR
dx dx
++=
+ + =
BY CHANGING THE DEPENDENT VARIABLE
AND REMOVING THE FIRST DERIVATIVE
REDUCTION TO NORMAL FORM
2
2
dy dy
PQyR
dx dx
++= (5.25)
Let
dy dv du
yuv v u
dx dx dx
=⇒ =+
(5.26)
22 2
22 2
2
d y d v du dv d u
uv
dx dx dx dx dx
=+ +
(5.27)
Substituting in (5.25) rearranging the terms
22
22
2
d v du dv d u du
uPu PQuvR
dx dx dx dx dx
⎛⎞
⎛⎞
++ + + + =
⎜⎟
⎜⎟
⎝⎠
⎝⎠
(5.28)
The term
dv
dx
will disappear if we choose such that
1
2
20i.e.
pdx
du
Pu x e
dx
+= =
(5.29)
2
2
2
11111
22242
du d u du dp dp
Pu P u P u u
dx dx dx dx dx
=− =− =
(5.30)
Substituting (5.29) and (5.30) into (5.28) w
e get
2
22
2
11 1
42 2
dv dp
uPuuPuQuvR
dx
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Publisher Resources

ISBN: 9781299487406Publisher Website