Wronskian | Mathematics Optional Notes for UPSC PDF Download

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5. Wronskian 
5.1 Find the Wronskian of the set of the function {?? ?? ?? ,|?? ?? ?? |} on the interval [-?? ,?? ] 
and determine whether the set is lirearly dependent on [-?? ,?? ]. 
(2009 : 12 Marks) 
Solution: 
Approach : lf functions are linearly dependent than the Wronskian as zero but the 
converse is not quite true as shown by these example. 
|3?? 3
|={
3?? 3
?? =0
-3?? 3
?? =0
?? ????
|3?? 3
|={
9?? 2
?? >0
-9?? 2
?? <0
 
? From ?? ?[-1,0) 
and for ?? ?(0,-??] 
 Wronskian =|
3?? 3
-3?? 3
9?? 2
-9?? 2
|=0 
 Wronskian =|
3?? 3
3?? 3
9?? 2
9?? 2
|=0 
i.e., Wronskian is identically zero on [-1,1]. 
But {3?? 3
,|3?? 3
|} are not linearly dependant on [-1,1]. 
Because if they were, ??? 1
,?? 2
 both not zero such that 
?? 1
3?? 3
+?? 2
|3?? 3
|=0 
?
 
?? 1
?? 2
=
-|3?? 3
|
3?? 3
={
1 ?? <0
-1 ?? -0
 So, ?? 1
=?? 2
,?? <0
 
??????                                                                                                ?? 1
=?? 2
,?? >0 
So, such a single value of ?? 1
 and ?? 2
 does not exist except if ?? 1
=?? 2
=0. 
?3?? 3
,|3?? 3
| are linearly independent.