# 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 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 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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