The general solution of the differential equation $\frac{x dy - y dx}{y} = 0$ is . . . . . . .

The general solution of the differential equation $x \cos y \,dy = (x e^x \log x + e^x) dx$ is given by

If $\frac{dy}{dx} = y + 3$ and $y(0) = 2$,then find the value of $y(\log 2)$.

The general solution of the differential equation $\frac{dy}{dx} = e^{x-y}$ is . . . . . . .

Find the equation of a curve passing through the point $(0, -2)$ given that at any point $(x, y)$ on the curve,the product of the slope of its tangent and the $y$-coordinate of the point is equal to the $x$-coordinate of the point.

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