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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 k_{1} and k_{2} both nonzero 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 k_{1} and k_{2} such that:
only when:
k_{1}=k_{2}=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:
k_{1}=1 and k_{2}=−1
Example 2:
Show that the functions
e^{t} and e^{2t}
are linearly independent on any interval.
For t∈I must show that
k_{1}=k_{2}=0
Choose two points t_{0}, t_{1}∈I where t_{0}≠t_{1}
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}=k_{2}=0
Theorem 3.3.1: If f and g are differentiable functions on an open interval I,
and
if for some point t_{0} 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 y_{1} and y_{2} 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 y_{1} and y_{2} 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 y_{1} and y_{2} be the solutions of Eq. (7)
where p and q are continuous on an open interval I. Then y_{1} and y_{2}
are linearly dependent on I if and only if is zero for all t in I .
Alternatively, y_{1} and y_{2} 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 y_{1}, y_{2} is given by
Equation 1 must be written in the form:
Equation 1 becomes :
Therefore:
and
Hence
It checks
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