mathematical method is part of the skill that u have a improve for the partial derivate equation

1. The Method Of Frobenius

2. Methods of Frobenius
• If x is not analytic, it is a singular point.
)()()()( '''
xfyxqyxpyxr =++
→
)(
)(
)(
)(
)(
)( '''
xr
xf
y
xr
xq
y
xr
xp
y =++
The points where r(x)=0 are called as singular points.

3. Methods of Frobenius (Cont’d)
• The solution for such an ODE is given as,
∑
∞
=
=
0m
m
m
r
xaxy
Substituting in the ODE for values of , )(),(),( '''
xyxyxy
equating the coefficient of xm
and obtaining the roots
gives the indical solution

4. .0atanalyticare)(and)(and
2
1
)(and,
2
3
)(
with
0)()(
aswrittenbecanequationThe
2
2
=
+
−==
=+′+′′
xxqxxxp
x
x
xq
x
xp
yxqyxpy
.0atpointsingularregularahas0)1(32equationThe 2
==+−′+′′ xyxyxyx
∑
∑
∑∑
∞
=
−+
∞
=
−+
∞
=
+
∞
=
+−+=′′
+=′
==
0
2
0
1
00
))(1()(
)()(
)(
n
sn
n
n
sn
n
n
sn
n
n
n
n
s
xAsnsnxy
xAsnxy
xAxAxxy

5. 0)(3)1)((2
gets,equationoriginaltheintoSubstitute
0
1
000
=−−++−++ ∑∑∑∑
∞
=
++
∞
=
+
∞
=
+
∞
=
+
n
sn
n
n
sn
n
n
sn
n
n
sn
n xAxAxAsnxAsnsn
∑∑
∞
=
+
−
∞
=
++
=
+=
1
1
0
1
Therefore,
1letbyindexofshiftamakeWe
n
sn
n
n
sn
n xAxA
nm
0)(3)1)((2
1
1
000
=−−++−++ ∑∑∑∑
∞
=
+
−
∞
=
+
∞
=
+
∞
=
+
n
sn
n
n
sn
n
n
sn
n
n
sn
n xAxAxAsnxAsnsn

6. ∑
∞
=
+
− =−−++−+++−+−
1
10 0}]1)(3)1)((2{[]13)1(2[
n
sn
nn
s
xAAsnsnsnxAsss
1or
0)1)(12(
or
013)1(2
then,0Since
22
1
1
0
−==
=+−
=−+−
≠
ss
ss
sss
A
1,]1)(3)1)((2[ 1 ≥=−++−++ − nAAsnsnsn nn

7. 1,
)32(
1
or
)32(
becomesthis,When
1
1
2
1
≥
+
=
=+
=
−
−
nA
nn
A
AAnn
s
nn
nn
0
012
01
)32...(75!
1
.
.
.
)75)(21(
1
72
1
51
1
Then
A
nn
A
AAA
AA
n
+•
=
••
=
•
=
•
=

8. 





+•
+=
=
∑
∞
=1
1
0
)32...(75!
1)(
solutionseriesgetwe,1Let
2
1
n
n
nn
x
xxy
A
1,
)32(
1
or
)32(
becomesrelationrecurrencethe,1When
1
1
≥
−
=
=−
−=
−
−
nA
nn
A
AAnn
s
nn
nn

9. 0
023
012
01
)32...(3(-1)1!
1
.
.
.
31)1(321
1
33
1
)1)(1(21
1
)1(2
1
)1(1
1
Then
A
nn
A
AAA
AAA
AA
n
−•
=
••−••
=
•
=
−•
=
•
=
−•
=

10. 





−•
−−=
=
∑
∞
=
−
2
1
2
0
)32...(31!
1)(
solutionseriessecondagetwe,1Let
n
n
nn
x
xxxy
A