How do you find singular solutions to differential equations? for example I am working with the equation $$\frac{dy}{dx}=y\cos^2(x)$$ This is a very easy separable equation. For some reason by intuition:
$$\frac{dy}{dx}=0 \to y\cos^2(x)=0$$
we have solutions at $y=0$ and $\cos^2(x)=0$
Am I on the right track. Is this justified how?
By separing variable $$\frac {dy}y= cos^2 x dx$$ Then integrating you obtain $$ ln |y| = \frac 12 (sen x cos x + x)$$ Finally you have $$y=\pm e^{\frac 12 (sen x cos x + x)}$$