first order differential equation from variational calculus problem has me stumped!

This arises when solving the problem of Dido,

$\displaystyle \frac{\dot x}{\sqrt{1+\dot x^2}} = -\frac{t}{\lambda} + C$, where C and $\displaystyle \lambda$ constant

They mention that it can be solved by direct integration or by using the substitution $\displaystyle \dot x =tan(\alpha)$.

The solution is given as circles of the form $\displaystyle (\frac{t}{\lambda}-d)^2+(\frac{t}{\lambda}-k)^2 = 1$ where d and k are constants of integration.

I don't know how to deal with that many differentials of x on the same side of the equation! I need some guidance here !