The solution of $\tan y \frac{dy}{dx} = \sin(x+y) + \sin(x-y)$ 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:

Let $y=y(x)$ be the solution of the differential equation $\frac{2+\sin x}{y+1} \cdot \frac{dy}{dx} = -\cos x$,where $y > 0$ and $y(0) = 1$. If $y(\pi) = a$ and $\frac{dy}{dx}$ at $x = \pi$ is $b$,then the ordered pair $(a, b)$ is equal to:

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

The general solution of $\frac{dy}{dx} + \sin \left(\frac{x+y}{2}\right) = \sin \left(\frac{x-y}{2}\right)$ is

The solution of the differential equation $ydx - xdy = x^2 ydx$ is