The solution of the differential equation $\frac{dy}{dx} = 1 + x + y + xy$ is

The solution curve of the differential equation $2 y \frac{dy}{dx} + 3 = 5 \frac{dy}{dx}$,passing through the point $(0, 1)$,is a conic whose vertex lies on the line:

The equation of the curve passing through the point $\left(0, \frac{\pi}{4}\right)$ and satisfying the differential equation $\left(e^x \tan y\right) dx + \left((1+e^x) \sec^2 y\right) dy = 0$ is given by

The general solution of the differential equation $\frac{dy}{dx} = 1 - x + y - xy$ is (where $C$ is a constant of integration)

The solution of $\frac{dy}{dx} = e^{-2x}$ with the condition $y(\log 2) = \frac{1}{16}$ is $y =$

The solution of $\frac{dy}{dx} = \frac{1}{x}$ is