5.6.7.8.

 
 
 
2
1
2
2
1 '' ' 0
'
'' ( 1)
( 1) 3 5 0
( 1) 3 5 0
2 5 0
1 2
1 2
cos(2ln ) sin(2
r
r
r
Given
Q a x y xy y
y x
y rx
y r r x
putting these inequation
r r r y
r r r
r r
r i complexconjugate roots
b general solution
roots are i
then general solution
A x B


   


 
   
   
  
  
 
  1ln )x x
ans
(2a)
   
 
 
 
 
   
 
 
 
 
0
0
0
0
1
2
1
1 1
0
0
2 '' 1 ' 0
1
1 1
1 int .
0
0
1 '
'' 0
, (1 )
0 sin int
1
1 1
1
Given
Q a xy x y xy
x
a x x
at x a x
x is ordinary po
x
a x x
a x
then
x y x
y y
x x
so p x x
q x x
hencebothq and p is analytic
so x is regular gular po
x
a x x
at x a x
x is ordina
   
 

   
 




  
 




   
  int .ry po
(2b)
 
   
 
 
 
 
2 2
2 2
0
0
2 2 2 2
1 2 2
1 2
1 1
1
2
2 (1 ) '' ' 0
1
1 ,0,1 sin int
0
2
'' ' 0
(1 ) (1 )
0
2
,
(1 )
(1 )
0 sin int
1
,
Given
Q b x x y y y
x
a x x x
x are gular po
a x
now
y y y
xx x x x
at x
so p x
x x
q x
x
hencebothq and p is not analytic
so x is irregular gular po
x
so p x
    
 
 


  
 







 
 
3
1 2
1 1
2
(1 )
)
(1 )
1 sin int
1 sin int
x x
x
q x
x x
hencebothq and p is analytic
so x is regular gular po
similarly x is regular gular po





 

 
 
 
 
    
  
2
2
1
1
0
2
0
4 2 '' ' (1 ) 0
0 sin int
1
'' 0
2
1
2
1
0 sin int
'' 1
2 1
n n
m r
m
m
m r
m
m
m r
m
Q x y xy x y
x is gular po
y x
y
x x
p x analytic
q x x analytic
so x is a refular gular po
i took a c
and n...