The Wronskian


Let tex2html_wrap_inline24 and tex2html_wrap_inline26 be two differentiable functions. We will say that tex2html_wrap_inline24 and tex2html_wrap_inline26 are proportional if and only if there exists a constant C such that tex2html_wrap_inline34 . Clearly any function is proportional to the zero-function. If the constant C is not important in nature and we are only interested into the proportionality of the two functions, then we would like to come up with an equivalent criteria. The following statements are equivalent:

Therefore, we have the following:

tex2html_wrap_inline24 and tex2html_wrap_inline26 are not proportional if, and only if, tex2html_wrap_inline52.

Define the Wronskian tex2html_wrap_inline54 of tex2html_wrap_inline24 and tex2html_wrap_inline26 to be tex2html_wrap_inline60 , that is

displaymath62

The following formula is very useful (see reduction of order technique):

displaymath64

Remark: Proportionality of two functions is equivalent to their linear dependence. Following the above discussion, we may use the Wronskian to determine the dependence or independence of two functions. In fact, the above discussion cannot be reproduced as is for more than two functions while the Wronskian does....


Mohamed Amine Khamsi
Wed Jul 17 17:22:52 MDT 1996

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