The solution of the equation $\frac{dy}{dx} = e^{x - y} + x^2 e^{-y}$ is

The solution of the differential equation $x \, dy - y \, dx = 0$ represents:

The solution of the differential equation $(x^2 + 1) \frac{dy}{dx} + (y^2 + 1) = 0$ is . . . . . .

If $y=y(x)$ is the solution of the differential equation $\left(\frac{2+\sin x}{y+1}\right) \frac{d y}{d x}+\cos x=0$ with $y(0)=1$, then $y\left(\frac{\pi}{2}\right)$ is equal to

The solution of the differential equation $\frac{dy}{dx} = (x+y)^2$ is

Find a particular solution of the differential equation $(x-y)(dx+dy)=dx-dy$ given that $y=-1$ when $x=0$. (Hint: put $x-y=t$)