Linear Equation of the Second Order with Variable Coeffi cients 5-3
Integrating we get
12
22
PdxPdx
Pdx
ee
VRuedxdxcdxc
uu
−−
⎡⎤
∫∫
∫
⎢⎥
=++
⎢⎥
⎣⎦
∫∫∫
Substituting this value of v in (5.2) we get
21
22
PdxPdx
Pdx
ee
ycucudxuRuedx
uu
−−
⎡⎤
∫∫
∫
⎢⎥
=++
⎢⎥
⎣⎦
∫∫∫
Since this includes the known solution ()yux= and it contains two
arbitrary constants, it is the general solution of (5.1).
5.1 TO FIND THE INTEGRAL IN C.F. BY INSPECTION,
i.e. TO FIND A SOLUTION OF
2
2
0
dydy
pQy
dxdx
++=
++=
2
2
0
dydy
PQy
dxdx
++=
(5.8)
1.
mx
ye=
is a solution of (5.8) if
2
0mpmQ++=
we have
2
2
2
,
mxmxmx
dydy
yememe
dxdx
===
If
mx
ye=
is a solution of (5.8) then
()
2
()0
mx
mPmQe++=
∴
2
0mPmQ++
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