The particular solution of the differential equation $$\left(1+y^2\right) \mathrm{d} x-x y \mathrm{~d} y=0$$ at $$x=1, y=0$$, represents

The general solution of the differential equation $$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{3 x+y}{x-y}$$ is (where $$C$$ is a constant of integration.)

The differential equation $$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\sqrt{1-y^2}}{y}$$ determines a family of circles with

The particular solution of the differential equation $$\left(1+e^{2 x}\right) d y+e^x\left(1+y^2\right) d x=0$$ at $$x=0$$ and y = 1 is