The general solution of the differential equation $e^{\frac{1}{2}\left(\frac{dy}{dx}\right)}=3^x$ is (where $C$ is a constant of integration).

The general solution of the differential equation $(2 y-1) dx - (2 x+3) dy = 0$ is

The particular solution of $\frac{y}{x} \frac{dy}{dx} = \frac{1+y^2}{1+x^2}$ when $x=2, y=1$ is

General solution of the differential equation $x \cos y \,dy = (x e^x \log x + e^x) dx$ is (where $C$ is a constant of integration.)

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 equation of the curve passing through $(3, 9)$ which satisfies the differential equation $\frac{dy}{dx} = x + \frac{1}{x^2}$ is