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In the book sadri hassani, the author proves that if the wronskian of 2 functions is zero, then the function is linearly dependent. The proof is:-

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I am not sure if the step where the $dx$ term is canceled is justified. One way to prove the same will be to rewrite the term as $\frac{d}{dx}(\frac{f_2}{f_1}) = 0$ and then prove it. Can someone shed some light on the method used by the author?

Rishabh Jain
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  • The proof is no different from the usual proof (which needn't be knowing the answer ahead of time). You have to separate variables at the last step and write $\dfrac{df_1}{f_1}=\dfrac{df_2}{f_2}$, getting $\log f_2 = \log f_1 + c$. – Ted Shifrin Feb 15 '20 at 17:56
  • Some people prefer the notation $df_1$ over $\frac{df_1}{dx}=\frac{df_1}{dx}(x)$ because $\frac{df_1}{dx}(x)=\frac{df_1}{dy}(y)=\frac{df_1}{dz}(z)=\dots$ (including the variable doesn’t contribute anything), but it basically means the same thing (slightly abusing notation). I think the author just computed $d(\frac{f_2}{f_1})$ as you mentioned, but I have no idea what else the author might be thinking – user651267 Feb 15 '20 at 17:56

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