Where Can I Find a Proof for the Wronskian Formula?

[zaybu]

Can anyone point to a proof for the Wronskian formula:

W[v,v*] = v'v* - vv*' = 2iIM(v', v*)

Thanks

 

[HallsofIvy]

What do you mean by 'IM(v', v*)'?

 

[ChrisVer]

$W= s z^{*}- s^{*}z$
$s=a+ib$
$z=c+id$

$W= (a+ib) (c-id) - (a-ib) (c+id)$
$W= ac -i ad +i bc -bd - ac -i ad +ibc+bd$
$W= -i ad +i bc -i ad +ibc$
$W= 2i bc - 2i ad = 2i (bc-ad)$

$sz^{*} = ac + bd + i (bc-ad)$
so
$Im(sz^{*})= bc-ad$

The proof is already seen

 

[zaybu]

Thanks

 

[blue_raver22]

for your question! The Wronskian formula is a mathematical tool used in the study of differential equations, and it is often used to determine whether a set of solutions to a differential equation is linearly independent or not. The formula is given by W[v,v*] = v'v* - vv*', where v and v* are two solutions to the differential equation and the prime notation (') represents differentiation with respect to the independent variable.

As for a proof of this formula, there are many resources available online that provide step-by-step explanations and derivations. One possible approach is to use the definition of linear independence and show that W[v,v*] = 0 if and only if the solutions v and v* are linearly dependent. Another approach is to use the property that the Wronskian of a set of solutions is a constant multiple of the determinant of the matrix formed by the solutions and their derivatives.

Ultimately, the proof relies on the properties of derivatives and linear independence, and it can be extended to more general cases involving higher order differential equations and multiple solutions. I hope this helps and encourages you to explore the fascinating world of differential equations and their applications further!