Wronskian: Solve "For W(x,fg,fh)=([f(x)]^2)W(g,h)

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Homework Help Overview

The discussion revolves around the Wronskian of functions, specifically the expression W(x, fg, fh) = ([f(x)]^2)W(g, h). Participants are exploring how to manipulate this expression using properties of determinants and derivatives.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster expresses confusion about applying the Wronskian in this context, particularly in relation to systems of equations. Some participants suggest using determinant properties and the product rule, while others question the notation used in the expression.

Discussion Status

Participants are actively discussing the manipulation of the Wronskian expression, with some guidance provided on using determinant properties. There is a clarification regarding the notation of the functions involved, indicating a productive exploration of the topic.

Contextual Notes

There is a noted uncertainty about the correct interpretation of the Wronskian notation, which may affect the discussion. The original poster's background in using the Wronskian with systems of equations contrasts with the current problem's requirements.

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"For the Wronskian, W, Show W(x,fg,fh)=([f(x)]^2)W(g,h)"

How is this done? I know how to use the Wronskian when there's a system of equations, something like y(x) = cosx, y(x)=sinx, y(x)=x, etc. But I'm really clueless about how to proceed here.
 
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W(x,fg,fh)=det([[fg,fh],[(fg)',(fh)']]). Right? Now use properties of the determinant. You can factor f out of the first row. Now you have f*det([[g,h],[(fg)',(fh)']]). Use the product rule on the second row. Now you can multiply any row of a determinant by a factor and add it to another row without changing the determinant. Also right? Add -f' times the first row to the second. Getting it yet?
 
is that supposed to say W(x,f(g),f(h)) on the left side?
 
Alex6200 said:
is that supposed to say W(x,f(g),f(h)) on the left side?

No. It's W(x,f(x)*g(x),f(x)*h(x)). Otherwise it doesn't work.
 

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