Conte Riccati and Jakob Hermann

[Poetria]

Homework Statement

Riccati sets y/x=q and then arrives at x^2*dq. This is his analysis of Jacob Hermann's differential equations criticised by Johannes Bernoulli (published in 1710).

x*dy-y*dx is a constant and is equivalent to dt.

I have understood everything except for the q-substitution.

The Attempt at a Solution

Well, I have tried several times but all my solutions are not correct. E.g. x*dx*(dq-q). I have no idea how he got this square. I am missing some clues.

 

[epenguin]

Perhaps you would find a more recent textbook easier to work from. Ideal is roughly 1900-1970, after that they get more difficult again.

If I have understood right your missing thing is the standard formula for derivative of a quotient, one of the half-dozen practically learned off by heart by most calculus students, see any calculus textbook:

y/x = q

dq = d(y/x) = (x dy - y dx)/x²

His formula Is just given by multiplying dq = (x dy - y dx)/x² by q²

 

[Poetria]

Perhaps you would find a more recent textbook easier to work from. Ideal is roughly 1900-1970, after that they get more difficult again.

If I have understood right your missing thing is the standard formula for derivative of a quotient, one of the half-dozen practically learned off by heart by most calculus students, see any calculus textbook:

y/x = q

dq = d(y/x) = (x dy - y dx)/x²

His formula Is just given by multiplying dq = (x dy - y dx)/x² by q²

Oh dear. What an idiot I am! Many thanks. :)

 

[epenguin]

Oh, I think when you have not done it for a year or two it fades. In fact, I am often not that sure whether to write x dy minus... or y dx minus... and have to stop and think about it.

I was intrigued by your avatar, guessed who she was though I did not remember the name amongst all the Madame de's offhand, and traced her via Voltaire.
I knew of Emilie du Chatelet's important translation of Newton, but I don't think I had known of her as the first to formulate of the law of conservation of energy.
You are doing some interesting studies.