Solve the differential equation $x\log x\ {\frac{d y}{d x}}+y\ =$${\frac{2}{x}}\mathrm{log}\,x.$

Find the particular solution of the following differential equation given that at x = 2, y = 1 $x\,{\frac{d y}{d x}}\,+\,2y=x^{2},\,(x\,\neq\,0).$

Find the particular solution of the differential equation : ${\frac{d y}{d x}}\ +y\cot x=2x+x^{2}\cot x,$ x ≠ 0. Given that y = 0, when $x={\frac{\pi}{2}}\ .$

Solve the following differential equation : ${\frac{d y}{d x}}\ +\,\sec x{y}=\tan x,\left(0\leq x<{\frac{\pi}{2}}\right)\!.$

Solve the following differential equation : (1 + x²) dy + 2xy dx = cot x dx, (x ≠ 0)

Find the particular solution of the following differential equation : ${\frac{d y}{d x}}-y=\cos x\ {\mathrm{~for~}}x=0,y=1.$

Solve the following differential equation : $x{\frac{d y}{d x}}\ +y-x+x y\cot x=0,x\neq0$

Find the particular solution of the differential equation : ${\frac{d y}{d x}}\ +y\;{\cot{x}}=$ 4x cosec x, (x ≠ 0), given, that y = 0,when $x={\frac{\pi}{2}}$

Solve the differential equation given as : $\left(\frac{e^{-2\sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right)\frac{d x}{d y}\;=1,x\neq0$

Solve $\left(1+x^{2}\right){\frac{d y}{d x}}+2x y-4x^{2}=0$ subject to the initial condition y(0) = 0.

Find the particular solution of differential equation : ${\frac{d y}{d x}}=-{\frac{x+y\cos x}{1+\sin x}}$ given that y = 1 when x = 0.

Find the general solution of the differential equation (1 + tan y)(dx – dy) + 2xdy = 0.

Find the particular solution of the differential equation : $y e^{y}\,d x=(y^{3}+2x e^{y})d y,y(0)=1$

Solve the differential equation $(\tan^{-1}x-y)d x=(1+x^{2})\,d y$

Find the general solution of the differential equation: ${\frac{d x}{d y}}\;=\;{\frac{y\tan y-x\tan y-x y}{y\tan y}}$