Obtain the differential equation of all the circles of radius r.

Solve the following differential equation. ${\sqrt{1+x^{2}+y^{2}+x^{2}y^{2}}}+x y{\frac{d y}{d x}}\,=0$

Solve the differential equation : ${\frac{d y}{d x}}-3y\cot x=\sin2x\operatorname{given}y=2,{\mathrm{~when~}}x={\frac{\pi}{2}}.$

Find the particular solution of the differential equation $\left(\tan^{-1}y-x\right)d y=(1+y^{2})\,d x,$ given that when x = 0, y = 0.

Find the particular solution of the differential equation (x – sin y) dy + (tan y) dx = 0, given that y = 0 when x = 0.

Find the particular solution of the differential equation $\left(1+x^{2}\right){\frac{d y}{d x}}=\left(e^{{\mathrm{mtan}^{\perp}x}}-y\right)$ given that y = 1 when x = 0.

Solve the differential equation $x\;\frac{d y}{d x}\;+y=$ x cos x + sin x, given that y = 1 when $x={\frac{\pi}{2}}$

Find the particular solution of the differential equation $(1+y^{2})+(x-e^{\mathrm{tan}^{-1}y})\ {\frac{d y}{d x}}\,=0,$ given that y = 0 when x = 1.