The integrating factor of the differential equation $(1 - x^2) \frac{dy}{dx} - xy = 1$ is

Let $y=y(x)$ be the solution of the differential equation $\operatorname{cosec}^{2} x \, dy + 2 \, dx = (1+y \cos 2x) \operatorname{cosec}^{2} x \, dx$,with $y(\frac{\pi}{4})=0$. Then,the value of $(y(0)+1)^{2}$ is equal to:

If $y(x)$ satisfies the differential equation $y^{\prime}-y \tan x=2 x \sec x$ and $y(0)=0$,then which of the following is true?

An integrating factor of the differential equation $\left(1-x^2\right) \frac{d y}{d x}+x y=\frac{x^4}{\left(1+x^5\right)}\left(\sqrt{1-x^2}\right)^3$ is

Let $f: R \rightarrow R$ be a continuous function which satisfies $f(x) = \int_{0}^{x} f(t) \, dt$. Then, the value of $f(\log_{e} 5)$ is:

The solution of the differential equation $x \frac{dy}{dx} + 2y = x^2$ $(x \neq 0)$ with $y(1) = 1$ is