Let the slope of the tangent line to a curve at any point $P(x, y)$ be given by $\frac{xy^2 + y}{x}$. If the curve intersects the line $x + 2y = 4$ at $x = -2$,then the value of $y$,for which the point $(3, y)$ lies on the curve,is ..... .

The solution of the differential equation $(x + 2y^3) \frac{dy}{dx} - y = 0$ is

Let $f:[0, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f(x) = 1 - 2x + \int_0^x e^{x-t} f(t) dt$ for all $x \in [0, \infty)$. Then the area of the region bounded by $y = f(x)$ and the coordinate axes is

The integrating factor of $\frac{dy}{dx} + y = \frac{1+y}{x}$ is

If the length of the sub-tangent at any point $P(x, y)$ on a curve $f(x, y) = 0$ is $x + 7y^2$, then $f(x, y) =$

Let a curve $y = y(x)$ pass through the point $(3,3)$ and the area of the region under this curve,above the $x$-axis and between the abscissae $3$ and $x (>3)$ be $\left(\frac{y}{x}\right)^{3}$. If this curve also passes through the point $(\alpha, 6\sqrt{10})$ in the first quadrant,then $\alpha$ is equal to $........$