Solution to second order second degree equation $ x^2 (d^2y/dx^2)^2=(dy/dx)^2+1$. I am unawae of any method that is there to solve such an equation. IT is totaly different from the usual linear differential equation of degree 2. Any Hint as to how to solve.

It certainly isn't linear. However, one thing we can do is reduce the order - let $u(x) = y' = \frac{dy}{dx}$, so that $u' = y''$. Then our equation becomes:
$$\begin{eqnarray}x^2u'^2 & = & u^2 + 1 \\
\frac{u'^2}{u^2 + 1} & = & \frac{1}{x^2} \\
\frac{u'}{\sqrt{u^2 + 1}} & = & \frac{1}{x} \end{eqnarray} $$
which is a separable first order equation (although you should also check the conditions, since I took the square root without checking whether to keep the $\pm$ sign in).
