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I am interested in solving the following nonlinear first-order differential equation that came out during a physics problem... The problem is that the derivative is a quadratic term, which I do not know how to deal with:

$$\biggl(\frac{dy}{dx}\biggl)^2 = a\frac{(y-b)(y-c)}{y^2}$$

where

a < 0

b, c > 0

b < y < c

I have tried to substitute and somehow become one derivative on each side of the equation, but that didn't work unfortunately...

Does anyone have any idea how this can be solved, or if it is at all solvable by analytic means?

Thank you!

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  • $\begingroup$ Take a square root $\endgroup$ – Yuriy S Dec 11 '17 at 18:02
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Take a square root of both sides, you get:

$$\frac{dy}{dx} = \pm \frac{\sqrt{a(y-b)(y-c)}}{y}$$

Separate the variables and integrate:

$$ \int \frac{ydy}{\sqrt{a(y-b)(y-c)}}= \pm x+C$$

The integral on the left hand side is simple: here it is

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