4

I was messing around with the Wronskian of two functions $y_1(x)$ and $y_2(x)$, which is defined by:

$$ W(y_1,y_2) = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix} = y_1y_2'-y_2y_1^{\prime} $$ And came across a supposedly new way of calculating the Wronskian, and I would like to know if it is correct: \begin{align} \mbox{We know that}\ \frac{d}{dx}\left(\frac{y_2}{y_1}\right) & = \frac{y_1y_2'-y_2y_1'}{y_1^2} \\[3mm] \mbox{So then:}\ \frac{W(y_1,y_2)}{y_1^2} & = \frac{d}{dx}\left(\frac{y_2}{y_1}\right) \\[3mm] \mbox{Which gives us:}\ W(y_1,y_2) & = y_1^2\left(\frac{d}{dx}\left(\frac{y_2}{y_1}\right)\right) \end{align} Will this always be true, assuming $y_1 \neq 0$ ?.

Sean Roberson
  • 10,119
cherrytree
  • 757
  • 1
    Well, yeah, you just showed it to be true. Are you asking if a similar formula holds for $3$ or more functions? – aqualubix May 31 '24 at 02:57
  • @aqualubix I was more wondering if I had done some things in the proof that were led by incorrect assumptions. I'm assuming this means that this is a valid way of calculating the Wronskian? – cherrytree May 31 '24 at 03:05
  • 1
    Sure. I don't see any mistake in your proof. It's an interesting formula, too. – aqualubix May 31 '24 at 03:06
  • I was also interested in this formula! Here are my posts on mathoverflow related to this, https://mathoverflow.net/questions/396458/connection-between-determinant-and-quotient-rule. https://mathoverflow.net/questions/396250/general-formulas-for-derivative-of-f-nx-dfracaxnbxn-1cxn-2-cdots – User May 31 '24 at 04:39

0 Answers0