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

[jaejoon89]

"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.

 

[Dick]

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?

 

[Alex6200]

is that supposed to say W(x,f(g),f(h)) on the left side?

 

[Dick]

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.