how to solve the differential equation $\sin^{2}(dy/dx) - x = 0$ for $y(0) = 0$ and what is its domain for this point?
Note: I have calculated it with online tools, so i am not asking the result. I am asking the steps for the result.
Note2: The question is true, it is exactly as i wrote it.( It is not like sin^2(x) etc.) No editing required.
By the non-negativity of the square, your equation is defined on $x\in [0,1]$ and can be simplified to $\cos(2y'(x))=1-2x$. Then with the inverse cosine $$ y'(x)=k\pi\pm\frac12\arccos(1-2x). $$ Each $k\in\Bbb Z$ gives a valid solution.