Differential equation $y[1+(y')^2]=c$ with $c$ an arbitrary constant, verifying solutions

Question

How can I verify that $$y= \frac{c}{2}(2\omega -\sin(2\omega) ) $$ and $$y= \frac{c}{2}(1 -\cos(2\omega) ) $$ are both solutions of the differential equation $$y[1+(y')^{2}]=c,$$ where c is an arbitrary constant. I'm just stuck at the replacement. I don't know what identities use to get c !!

Answer 1

For the second solution, you have: $$y= \frac{c}{2}(1 -\cos(2\omega) )$$ $$\implies y'=c \sin (2\omega)$$ Plug that in the equation and see if the equality holds. $$y[1+(y')^{2}]=c,$$ $$(1 -\cos (2\omega) )(1+c^2 \sin ^2 (2\omega))=2$$ $$.........$$ This must hold for any $\omega$. On the other hand, you should have a different constant for the solution $y(x)$. C is already in the differential equation.