Q.Solve $\displaystyle\frac{dy}{dx} = 2xy$ given that $y = 1$ when $x = 0$.

A first-order equation of the form $\frac{dy}{dx}=f(x)g(y)$ is solved by separating the variables to $\frac{dy}{g(y)}=f(x)\,dx$ and integrating each side, adding a single arbitrary constant. For instance $\frac{dy}{dx}=\frac{x}{y}$ gives $y\,dy=x\,dx$, so $\frac{y^2}{2}=\frac{x^2}{2}+c_1$, i.e. $y^2-x^2=c$. A general s …