MHB Please help me solve this differential equation for the initial condition (0,-1)

[hornekat]

Please help me solve this differential equation for the initial condition (0,-1):
dx/dy = ((1+x^2)^(1/2))/(xy^3)

I think I'm doing something wrong because I end up with
((x^2)(y^3))/2 = ((x^2)+y)^(1/2) + c,
but when plugging in the initial condition it ends up being the square root of negative one.

Please help, thank you!

 

[MarkFL]

The ODE is:

$$\d{x}{y}=\frac{\left(x^2+1\right)^{\frac{1}{2}}}{xy^3}$$

Separating variables, we obtain:

$$\frac{x}{\left(x^2+1\right)^{\frac{1}{2}}}\,dx=y^{-3}\,dy$$

And now integrating, we get:

$$\left(x^2+1\right)^{\frac{1}{2}}=-\frac{1}{2}y^{-2}+C$$