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

If $\frac{dy}{dx} = 1 + x + y + xy$ and $y(-1) = 0$,then the function $y$ is

If the differential equation $\frac{dy}{dx} + \frac{x}{y} = \frac{a}{y}$,where $a$ is a constant,represents a family of circles,then the radius of the circle is ......

The solution of the differential equation $\frac{dy}{dx} = x^2 + \sin 3x$ is

Let $y=f(x)$ be the solution of the differential equation $y(x+1) dx - x^2 dy = 0$ with the initial condition $y(1)=e$. Then $\lim _{x \rightarrow 0^{+}} f(x)$ is equal to

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