The solution of the differential equation $\frac{dy}{dx} = \sec x(\sec x + \tan x)$ is

At any point on a curve,the slope of the tangent is equal to the sum of the abscissa and the product of the ordinate and abscissa of that point. If the curve passes through $(0, 1)$,then the equation of the curve is

Let $y=y(x)$ be the solution of the differential equation $x^3 dy + (xy - 1) dx = 0, x > 0$,with $y(\frac{1}{2}) = 3 - e$. Then $y(1)$ is equal to

Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a differentiable function. If $6 \int_{1}^{x} f(t) dt = 3xf(x) + x^{3} - 4$ for all $x \ge 1$,then the value of $f(2) - f(3)$ is

The solution of the differential equation $\frac{dy}{dx} - 2y \tan 2x = e^x \sec 2x$ is

$A$ family of curves has the differential equation $x y \frac{d y}{d x}=2 y^2-x^2$. Then,the family of curves is