Skip to Main Content
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
5-8 Diff erential Equations
Substituting
yvx=
and hence
dy dv
vx
dx dx
=+
and
22
22
2
dy dv dv
x
dx dx dx
=+
The given differential equation becomes
()
2
1
2
2
22
2
1
1
dv x dv
x
dx x x dx
⎛⎞
++ =−
⎜⎟
⎝⎠
Substituting p for
dv
dx
this becomes
()
1
2
2
2
2
1
1
dp x
px
dx x x
⎛⎞
++ =−
⎜⎟
⎝⎠
This is a linear equation with integrating factor
()
2
1
2
2
1
x
x
Its solution is
()
2
3
1
1
2
2
1
3
1
x
pxc
x
=+
or
1
22
2
1
11
3
c
dv
pxx x
dx x
== −+
Integrating again we get
()
2
3
21
2
12
11
1sin
9
x
vxcx c
x
⎛⎞
=− + + +
⎜⎟
⎝⎠
Hence the complete solution is
()
()
3
212
2
12
1
1sin1
2
yvx x x cx x x cx
==− + +− +
Example 5.1.6 Solve
(
)
2
2
sin cos cos cos 0
dy dy
xx x x x y x
dx dx
+−+=
given that yx= is a solution.
Solution Substituting
yvx=
, we get
2
2
22 ...
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.
Start your free trial

You might also like

Differential Equations

Differential Equations

Marian Mureşan
Engineering Mathematics

Engineering Mathematics

P. Sivaramakrishna Das, C. Vijayakumari

Publisher Resources

ISBN: 9781299487406Publisher Website