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

The general solution of the differential equation $\frac{dy}{dx} = \cot x \cdot \cot y$ is

If $y = y(x)$ is the solution of the differential equation $(1 + e^{2x}) \frac{dy}{dx} + 2(1 + y^2)e^x = 0$ and $y(0) = 0$,then $6(y'(0) + (y(\log_e \sqrt{3}))^2)$ is equal to

The solution of $\log \left( \frac{dy}{dx} \right) = ax + by$ is

The general solution of the differential equation $\cos(x+y) dy = dx$ is

The particular solution of the differential equation $y(1+\log x) = (\log x^x) \frac{dy}{dx}$,given $y(e) = e^2$,is