Is Separation of Variables possible here?

Question

I was asked to solve the differential equation $$\frac{dy}{dx}=1+y^{2}.$$ The issue I'm finding is that each attempt to isolate $y$ adds a $dx$ next to it. For example, if I multiply both sides by $dx$, I get $$dy = dx-y^{2}dx,$$ and subtracting $y^{2}dx$ yields $$y^{2}dx-dy=dx.$$ This is technically a step closer to complete separation of variables (all $y$ variables are on the same side), but the $dx$ next to $y^{2}$ problematizes things. To illustrate this, dividing $dx$ yields $$y^{2}-\frac{dy}{dx}=1,$$ essentially right where I started. Is there a rule regarding Separation of Variables I'm missing, or is the technique not viable for this Differential Equation?

Answer 1

Yes of course

$$\frac{dy}{dx}=1+y^{2}\implies\frac{dy}{1+y^2}=dx\implies \arctan y=x+c$$