Linear Independence and the Wronskian

Let tex2html_wrap_inline41 and tex2html_wrap_inline43 be two differentiable functions. The Wronskian tex2html_wrap_inline45 , associated to tex2html_wrap_inline41 and tex2html_wrap_inline43 , is the function

displaymath33

For a discussion on the motivation behind the Wronskian, click HERE.

We have the following important properties:

(1)
If tex2html_wrap_inline41 and tex2html_wrap_inline43 are two solutions of the equation y'' + p(x)y' + q(x)y = 0, then

displaymath34

(2)
If tex2html_wrap_inline41 and tex2html_wrap_inline43 are two solutions of the equation y'' + p(x)y' + q(x)y = 0, then

displaymath35

In this case, we say that tex2html_wrap_inline41 and tex2html_wrap_inline43 are linearly independent.

(3)
If tex2html_wrap_inline41 and tex2html_wrap_inline43 are two linearly independent solutions of the equation y'' + p(x)y' + q(x)y = 0, then any solution y is given by

displaymath36

for some constant tex2html_wrap_inline77 and tex2html_wrap_inline79 . In this case, the set tex2html_wrap_inline81 is called the fundamental set of solutions.

Example: Let tex2html_wrap_inline41 be the solution to the IVP

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and tex2html_wrap_inline43 be the solution to the IVP

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Find the Wronskian of tex2html_wrap_inline81 . Deduce the general solution to

displaymath93

Solution: Let us write tex2html_wrap_inline95 . We know from the properties that

\begin{displaymath}W(x) = W(0) e^{\displaystyle - \int_0^x (2t-1)dt} = W(0) e^{\displaystyle -x^2 + x}\;.\end{displaymath}

Let us evaluate W(0). We have

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Therefore, we have

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Since tex2html_wrap_inline105 , we deduce that tex2html_wrap_inline81 is a fundamental set of solutions. Therefore, the general solution is given by

displaymath36,

where tex2html_wrap_inline111 are arbitrary constants.

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Author: Mohamed Amine Khamsi

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