During my study of calculus of variations I came across a nonlinear DE. I know that its solution is hyperbolic cosine with some constants yet do not know how to proceed since the function of interest is inside a square root. Also I am unfamiliar with the procedure of solving nonlinear differential equations. The equations is as follows: $$\frac{dy}{dx}=\pm\sqrt{\frac{\sigma g y + \lambda}{C} - 1} $$
EDIT: This equation is got after we try to minimize the potential energy of a cable hung between two poles with a constraint on its length a uniform linear mass density of $\sigma$ and $\lambda$ is a Lagrange multiplier.
Are you sure the solution is a combination of hyperbolic cosines?
Let's see. Re-write the ODE as follows:
$$ \pm \frac{dy}{\sqrt{k y - \beta}} = dx,$$
where $k = \sigma g /C$ and $\beta = 1-\lambda/C$. The equation above is equivalent to:
$$ \pm \frac{dy}{\beta^2 \sqrt{\frac{ky}{\beta} - 1}} = dx,$$
define $z:= ky/\beta$, so $dy = \beta dz/k $ and the differential equation becomes:
$$ \pm \frac{dz}{k \beta \sqrt{z-1}} = dx,$$
so a first integration gives us:
$$ \pm \frac{2 \sqrt{z-1}}{k \beta} = x + A,$$
being $A$ a constant of integration. Now you can explicitly solve for $z = z(x)$ as follows:
$$z(x) = 1+ \left( \frac{k\beta}{2} \right)^2 \left(x+A\right)^2.$$
You can obtain the solution for $y(x)$ in terms of the change of variable carried out before.
Cheers!