The general solution of the differential equation $(2x - 10y^3) dy + y dx = 0, y \neq 0$ is

Let $f$ be a real-valued differentiable function on $\mathbb{R}$ (the set of all real numbers) such that $f(1)=1$. If the $y$-intercept of the tangent at any point $P(x, y)$ on the curve $y=f(x)$ is equal to the cube of the abscissa of $P$,then the value of $f(-3)$ is equal to

Let $y = y(x)$ be the solution of the differential equation $\frac{dy}{dx} + y \tan x = 2x + x^2 \tan x$,$x \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$,such that $y(0) = 1$. Then

The general solution of the differential equation $(y^2+x+1) dy = (y+1) dx$ is

If $x^2 y - x^3 \frac{dy}{dx} = y^4 \cos x$,then $x^3 y^{-3}$ is equal to

Let $y=y(x)$ be a solution curve of the differential equation $(y+1) \tan ^{2} x \,dx+\tan x \,dy+y \,dx=0$ for $x \in \left(0, \frac{\pi}{2}\right)$. If $\lim _{x \rightarrow 0+} x y(x)=1$,then the value of $y\left(\frac{\pi}{4}\right)$ is: