- #1

Portuga

- 56

- 6

- Homework Statement
- Solving a Brachistochrone curve with drag.

- Relevant Equations
- ##\vec F = -k \vec v##, ##v = \frac{C}{\sqrt{1 + \left( y^\prime \right)^2}}##

Well, I followed the strategy used by A.S. Parnovsky in his article (\url{http://info.ifpan.edu.pl/firststep/aw-works/fsV/parnovsky/parnovsky.pdf}) and found this differential equation: $$-\frac{g x}{C^{2}} = -\frac{\beta^{2} {y^{\prime}}^{2} \arctan\left({y^{\prime}}\right) + \beta {y^{\prime}}^{2} \log\left({y^{\prime}}^{2} + 1\right) - 2 \, \beta {y^{\prime}}^{2} \log\left(\beta - {y^{\prime}}\right) - \beta^{3} - \beta^{2} {y^{\prime}} + \beta^{2} \arctan\left({y^{\prime}}\right) - {y^{\prime}}^{2} \arctan\left({y^{\prime}}\right) + \beta \log\left({y^{\prime}}^{2} + 1\right) - 2 \, \beta \log\left(\beta - {y^{\prime}}\right) - \beta - {y^{\prime}} - \arctan\left({y^{\prime}}\right)}{2 \, {\left(\beta^{2} + 1\right)}^{2} {\left({y^{\prime}}^{2} + 1\right)}},$$ where ##\beta, g## and ##C## are constants.

The problem is: how to proceed? This is clearly an transcendental equation for ##y^\prime## and the autor didn't solve it explicitly. He purposed the use of a series expansion of reciprocal powers, but I was imagining if this was the most reasonable way to do it.

The problem is: how to proceed? This is clearly an transcendental equation for ##y^\prime## and the autor didn't solve it explicitly. He purposed the use of a series expansion of reciprocal powers, but I was imagining if this was the most reasonable way to do it.