English | Español

Try our Free Online Math Solver!

Online Math Solver

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


Linear Independence and the Wronskian

Two functions , f and g are said to be linearly dependent on the interval I if

there exists two constants k1 and k2 both non-zero such that:

(For all t that exist in the interval I)

Two functions , f and g are said to be linearly independent on the interval I if

there exists two constants k1 and k2 such that:



only when:

k1=k2=0

Example 1:

Is   linearly independent or linearly dependent?

The given functions are linearly dependent on any interval, since:



for all t if we chose:

k1=1 and k2=−1

Example 2:

Show that the functions

et and e2t

are linearly independent on any interval.
For t∈I must show that

k1=k2=0

Choose two points t0, t1∈I where t0≠t1
We suppose that



Evaluate this equation at these points.



Evaluate the determinant of coefficients



Since the determinant is not zero, the only possible solution is

k
1=k2=0

Theorem 3.3.1: If f and g are differentiable functions on an open interval I, and
if for some point t0 in I, then f and g are linearly
independent on I. Moreover, if f and g are linearly dependent on I, then
for every t in I.

Theorem 3.3.2: (Abel's Theorem) If y1 and y2 are solutions of the differential
equation



where p and q are continuous on an open interval I, then the Wronskian
is given by:



where c is a certain constant that depends on y1 and y2 but not on t.
Further, either is zero for all t in I (if c=0 ) or else is never
zero in I (if c≠0 ).

Theorem 3.3.3: Let y1 and y2 be the solutions of Eq. (7)



where p and q are continuous on an open interval I. Then y1 and y2
are linearly dependent on I if and only if is zero for all t in I .
Alternatively, y1 and y2 are linearly independent on I if and only if
is never zero in I .

Example 3:

Given



it was verified that



Verify that the Wronskian of y1, y2 is given by



Equation 1 must be written in the form:



Equation 1 becomes :



Therefore:

and

Hence

It checks

Prev Next