The solution of the differential equation $\frac{dy}{dx} = x^2 + \sin 3x$ is

The solution of $x^2 + y^2 \frac{dy}{dx} = 4$ is

The general solution of the differential equation $y \log y \, dx - x \, dy = 0$ is . . . . . . .

Let $\alpha |x| = |y| e^{xy-\beta}$,where $\alpha, \beta \in \mathbb{N}$,be the solution of the differential equation $x dy - y dx + xy(x dy + y dx) = 0$ with the initial condition $y(1) = 2$. Then $\alpha + \beta$ is equal to:

The particular solution of the differential equation $y(1+\log x) \frac{dx}{dy} - x \log x = 0$ when $x=e, y=e^2$ is

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