Suppose I have the first-order nonlinear ODE: $$(y’)^2+y^2=1$$
Upon inspection, we see that $y(t)=\pm1$ and $y(t)=\pm\sin(t+a)$. ‘Another’ solution one sees upon inspection is $y(t)=\pm\cos(t+a),$ but this is contained in the other solution ($\pm\sin(t+a)$).
My question is:
Are these solutions unique? Are there any others and how do we know? Thanks in advance.