The equation of the curve whose slope at any point is equal to $2xy$ and which passes through the point $(0,1)$ is

The solution of the differential equation $(1 + x^2)\frac{dy}{dx} = x(1 + y^2)$ is

The solution of $\frac{dy}{dx} = \sqrt{1-y^2}$ with the initial condition $y(0) = 1$ is:

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 $y(x)$ of the differential equation $\frac{d}{dy} \left( \int_x^y dt \right) = x$ is:

General solution of the differential equation $\frac{dy}{dx} = \frac{x+y+1}{x+y-1}$ is given by