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

The solution of $\cos y \frac{dy}{dx} = e^{x+\sin y} + x^2 e^{\sin y}$ is $f(x) + e^{-\sin y} = C$ ($C$ is an arbitrary real constant), where $f(x)$ 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 differential equation $x^2(y+1) dx + y^2(x-1) dy = 0$ has the general solution given by (where $C$ is a constant of integration.)

The differential equation $\cos (x+y) dy = dx$ has the general solution given by

The particular solution of the differential equation $\frac{dy}{dx} = e^{2y} \cos x$,when $y(\frac{\pi}{6}) = 0$ is