How is this differential equation solved? I have a differential equation as follows:
$$ y \cdot y'' +(y')^2 +1=0.$$
I'm interested in how to solve it. So far I have found a few solutions like $y=\sqrt{r^2-(x-k)^2}$ and $y=x+k.$ [In these, $r$ and $k$ are real constants.]
It came up while looking at a property of surface area of revolutions which held for spheres and certain cones. I am wondering if a general soution can be found, and in particular whether only circles and certain lines are solutions.
Note: As user RadialArmSaw noted, $y=x+k$ is not a solution.

Noting that $y y'' + (y')^2 + 1 = (y y' + x)'$, the original equation reduces to $y y' + x = c_1$, which is an equation with separable variables whose solution is implicitly defined by the relation
$$
\frac{y^2}{2} = c_1 x-\frac{x^2}{2}+c_2
$$
that can be rewritten as
$$
y^2=k_1 x-x^2+k_2 \Leftrightarrow (x-\alpha_1)^2+y^2 = \alpha_2.
$$
Given initial or boundary conditions, we can compute (when possible) the constants $\alpha_1, \alpha_2$. When the initial/boundary data is compatible with the differential equation the solution is a circumference, as noted by others.
