3.3 Linear Independence and Wronskian
Definition:
Two functions f (t ) and g (t ) are said to be
linearly dependent on an interval I if there exist
two constants k 1 and k 2 not both zero, such that
k 1 f (t )  k 2 g (t )  0
(1)
For all t  I .
The functions f (t ) and g (t ) are said to be linearly
independent on an interval I if they are not
linearly dependent.
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Theorem 3.3.1
If f (t ) and g (t ) are differentiable functions on
an open interval I and if W (f , g )(t 0 )  0 for some
point t 0  I then f (t ) and g (t ) are linearly
independent on the interval I .
Moreover, if f (t ) and g (t ) are linearly dependent
on an interval I then W (f , g )(t )  0 for every
t I .
2
Theorem 3.3.2 (Abel's Theorem)
If y 1 and y 2 are two solutions of the DE
y   p (t ) y   q (t ) y  0
(2)
where p (t ) and q (t ) are continuous on an open
interval I , then the Wronskian W ( y 1 , y 1 )(t ) is
given by
W ( y 1 , y 1 )(t )  c exp   p (t ) dt 
(3)
Where c is a constant that depends on y 1 and
y 2 but not on t .
Note:
 0,
W ( y 1 , y 1 )(t ) 
 0,
3
c  0,
c  0.
1
2
1
Example 1: Assume that y 1  t and y 2  t are
solution of the DE
2t 2 y   3ty   y  0, t  0.
Use Abel's Theorem to find the constant c in (3).
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Theorem 3.3.3:
Let y 1 and y 2 be solutions of the DE
y   p (t ) y   q (t ) y  0
where p (t ) and q (t ) are continuous on an open
interval
I,
then
y 1 and
y2
are linearly
dependent on I if and only if W ( y 1 , y 1 )(t ) is zero
for all t  I .
Alternatively, y 1 and y 2 are linearly independent
on I if and only if W ( y 1 , y 1 )(t ) is never zero for
all t  I .
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Summary: We can summarize the facts about
fundamental set of solutions, Wronskian and
linear independence in the following:
Let y 1 and y 2 be solutions of the DE
y   p (t ) y   q (t ) y  0
where p (t ) and q (t ) are continuous on an open
interval I . Then the following statements are
equivalent.
1- The
functions
y1
and
y2
are
a
fundamental set of solutions on I .
2- The functions y 1 and
y 2 are lineary
independent on I .
3- W ( y 1 , y 1 )(t 0 )  0 for some t 0  I .
4- W ( y 1 , y 1 )(t )  0 for all t  I .
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H.W. 1-13, 15-23 Page 158
Example 2(Q.4, page 158): Determine whether
the following functions are linearly independent
or linearly dependent
f (x )  e 3x and g (x )  e 3( x 1)
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Example 3(Q.18, page 158) Find the Wronskian
of two solutions of the given DE
Example 4(Q.4, page 158)
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