If $y=y(x)$ and $\frac{2+\sin x}{y+1}\left(\frac{d y}{d x}\right)=-\cos x$,with $y(0)=1$,then $y\left(\frac{\pi}{2}\right)$ 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

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

Let $y = y(x)$ be the solution of the differential equation $\frac{dy}{dx} = (1+x^2)(1-y^2)$, with the initial condition $y(0) = \frac{1}{2}$. Then the value of $(2y(1) - 1)$ is equal to:

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

Let $y=y(x)$ be a solution curve of the differential equation,$(1-x^2 y^2) dx = y dx + x dy$. If the line $x = 1$ intersects the curve $y = y(x)$ at $y = 2$ and the line $x = 2$ intersects the curve $y = y(x)$ at $y = \alpha$,then a value of $\alpha$ is