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

Let $y(x)$ be a solution of $\frac{(2 + \sin x) dy}{(1 + y) dx} = \cos x.$ If $y(0) = 2,$ then $y\left( \frac{\pi}{2} \right)$ equals

The general solution of ${x^2}\frac{{dy}}{{dx}} = 2$ is

The particular solution of the differential equation $(y + x \cdot \frac{dy}{dx}) \cdot \sin(xy) = \cos x$ at $x = 0$ is

The solution of $(x\sqrt{1 + y^2})dx + (y\sqrt{1 + x^2})dy = 0$ is

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