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This is a part of a question on the master/doctor degree graduate school entrance test for Tokyo Univ. in 2016.

The conditions are below

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Here is the question.

I have to make a differential equation using y, y', and c.

However, I don't have any ideas to partially derivate F with respect to y'.

I can't ignore y in F(y,y') because it has a relationship with y, can I?

Please tell me how to deal with this problem.

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The partial derivative of F[y,y'] is calculated by treating y as a constant and taking the derivative relative to y'. In your case:

D[y Sqrt[1 + y' ^2], y']

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  • $\begingroup$ But if y=(x^2)/2 and y' = x, F((x^2)/2,x) = x^2*sqrt(1+x^2)/2. In this case, I can't ignore x^2 because It can be derivated with respect to x, can I? $\endgroup$ – Kanon Mar 11 at 3:28
  • $\begingroup$ No, if you say dF[y,y']/dy' you mean: You treat y and y' as the only variables, independent of what y means. $\endgroup$ – Daniel Huber Mar 11 at 8:38

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