How to solve this problem using simpliest differential equation

Question

Find all curves, that the length of the segment between the touch point and the intersection point of the tangent at the given point with the axis OX, the constant.

It should be easy. I thought about this: $\frac{x^2}{(x')^2} + x^2 = \text{const}^2$ (easy because of the Pythagorean theorem), but I am not aware of solving that..I don't know any methods except separation of variables. Can you help me with that?

Answer 1

The differential equation being $$\frac{x^2}{(x')^2} + x^2 = c^2$$ with $x=x(t)$, you can rewrite it as $$x^2(t')^2+x^2=c^2$$ where $t=t(x)$ that is to say $$t'=\pm \frac{\sqrt{c^2-x^2}}{x}$$ in which the variables have been separated. Integrating both sides will give $$t+C=\pm \int \frac{\sqrt{c^2-x^2}}{x}\,dx$$ leading to a implicit equation (which, I suppose, will not allow to express $x(t)$).