The equation of the curve passing through the origin and satisfying the differential equation $(1+x^2) \frac{dy}{dx} + 2xy = 4x^2$ is:

Let $y=y(x)$ be the solution of the differential equation $(1-x^2) dy = [xy + (x^3+2) \sqrt{3(1-x^2)}] dx$ for $-1 < x < 1$,with $y(0)=0$. If $y(1/2) = m/n$,where $m$ and $n$ are coprime numbers,then $m+n$ is equal to . . . . . . . . . .

If the curve $y=y(x)$ is the solution of the differential equation $2(x^{2}+x^{5/4}) dy - y(x+x^{1/4}) dx = 2x^{9/4} dx, x > 0$ which passes through the point $(1, 1-\frac{4}{3} \log_{e} 2)$,then the value of $y(16)$ is equal to :

If the general solution of the differential equation $\cos^2 x \frac{dy}{dx} + y = \tan x$ is $y = \tan x - 1 + Ce^{-\tan x}$ and it satisfies $y(\frac{\pi}{4}) = 1$,then $C =$

If the function $y = f(x)$ satisfies the differential equation $(x^3 + 1)dy = x(1 - 3xy)dx$ and $f(0) = 0$,then $\mathop {\lim }\limits_{x \to 0} \frac{x^2}{f(x)}$ is equal to

The integrating factor of the differential equation $(2x + 3y^2) dy = y dx$ $(y > 0)$ is