# Solving differential equation: $\frac{dx}{dy}=\frac{G(y)}{\sqrt{1-G^2(y)}}$

I'm carrying out an experiment in which I have to solve the differential equations:

$$\dfrac{dx}{dy}=\dfrac{A\left(ly-\frac{y^2}{2}\right)}{\sqrt{1-A^2\left(ly-\frac{y^2}{2}\right)^2}}.$$

I don't know if one can do it by integration (I couldn't), but Wolfram Alpha doesn't come up with any solution. I can solve it numerically. I have some data $\{(x,y)\}$ and I would like to fit the equation $x(y)$ to get the constant $A$. I already know this constant experimentally, but I'd like to get a computational value for it. That's why I need the analytical solution. Thanks for your help

• Not to say there isn't possibly an analytical solution. But if WolframAlpha can't find one, then it is unlikely to exist. – Simon S May 17 '15 at 16:24
• If only there was the term $(y-l)$ in the numerator, then the integration would be straightforward by reversing the chain rule, leading to $x=\sqrt{1-A^2(ly-\frac{y^2}{2})}+C$. But, in its present form, it doesn't seem like an analytical solution is possible. – Alijah Ahmed May 17 '15 at 16:32
• @Vladimir Vargas: In the equation, they are two parameters $A$ and $l$. Are they both unknown ? I mean : Is a fitting problem of non-linear regression for one or two parameters ? – JJacquelin May 9 '17 at 6:13
• @Vladimir Vargas: Could you edit an example of data ${(x,y)}$ in order to test a method of regression. – JJacquelin May 9 '17 at 9:11

Analytical solving is possible. The result can be expressed on the form of the inverse function $x(y)$. The resulting equation involves elliptic integrals of the first and second kind (see below). That is complicated and moreover, the equation cannot be inverted on a closed form for $y(x)$.
So, I think that the analytic solution is of no help to fit the equation $x(y)$ to the data {(x,y)}. The fitting might be possible thank to a method of nonlinear regression.