Linear Equation of the Second Order with Variable Coeffi cients 5-13
which is a linear equation in p. The integrating factor is
2
11
2log
2log
2
1
dx
x
xx
eee
x
−
−
∫
===
Its solution is
1
22
112
1
x
pedxc
xxx
⎛⎞
=−+
⎜⎟
⎝⎠
∫
()
23
1
2
xx
xexedxc
−−
=−+
∫
(
)
(
)
(
)
2
1
xxx
xecefxfxdxefx
−
=+∴+=
′
⎡⎤
⎣⎦
∫
1
x
dv
pecx
dx
==+
Integrating
3
12
3
x
x
vecc
⎛⎞
=++
⎜⎟
⎝⎠
The general solution of (5.12) is
3
2
12
3
xxxx
x
yvececee==++
Example 5.1.11 Solve
2
2
2
(2)2.
dydy
xxyx
dxdx
−−=
Solution Dividing by x, the giv
en equation in the standard form is
2
2
2
22
1
dydy
yx
dxxdxx
⎛⎞
+−−=
⎜⎟
⎝⎠
(5.16)
where
2
22
1,,PQRx
xx
=−=−=
Since
22
1110PQ
xx
−+=−+−=
x
ye
−
=
is a part of the C. F. of (5.16)
Let
x
yve
−
=
, where
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