The particular solution of the diffrential equation $$y(1+\log x)=\left(\log x^x\right) \frac{d y}{d x}$$, when $$y(e)=e^2$$ is

The general solution of $$\sin ^{-1}\left(\frac{d y}{d x}\right)=x+y$$ is

Solution of the differential equation $$\mathrm{y'=\frac{(x^2+y^2)}{xy}}$$, where y(1) = $$-$$2 is given by

The differential equation of all family of lines $$y=m x+\frac{4}{m}$$ obtained by eliminating the arbitrary constant $$\mathrm{m}$$ is