An integrating factor of the differential equation $\frac{dy}{dx} + \frac{2xy}{1 - x^2} = \frac{x}{\sqrt{1 - x^2}}$ is

Let $f: [1, \infty) \rightarrow R$ be a differentiable function such that $f(1) = \frac{1}{3}$ and $3 \int_1^x f(t) dt = x f(x) - \frac{x^3}{3}$ for $x \in [1, \infty)$. Then the value of $f(e)$ is:

The solution of the differential equation $x \cos x \frac{dy}{dx} + (x \sin x + \cos x) y = 1$ is

If $\frac{dy}{dx} + 2y \tan x = \sin x$,$0 < x < \frac{\pi}{2}$ and $y(\frac{\pi}{3}) = 0$,then the maximum value of $y(x)$ is.

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

Let $y=f(x)$ be the solution of the differential equation $\frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^6+4x}{\sqrt{1-x^2}}$ for $-1 < x < 1$ such that $f(0)=0$. If $6 \int_{-1/2}^{1/2} f(x) dx = 2\pi - \alpha$,then $\alpha^2$ is equal to . . . . . .