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

$A$ continuous function $f: R \rightarrow R$ satisfies the equation $f(x) = x + \int_0^x f(t) \, dt$. Which of the following options is true?

Let $f : (0, \infty) \to (2, 20)$ be a twice differentiable function such that $\lim_{x \to \infty} (f(x) + f'(x) + f''(x)) = \lim_{x \to \infty} g(x)$,where $\lim_{x \to \infty} g(x)$ exists and is equal to $5$. Then $\lim_{x \to \infty} (f(x) - g(x))$ is equal to:

The solution of the differential equation $(x + \log y)dy + y\,dx = 0$ is:

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

Let $\alpha$ be a non-zero real number. Suppose $f: R \rightarrow R$ is a differentiable function such that $f(0)=2$ and $\lim _{x \rightarrow-\infty} f(x)=1$. If $f^{\prime}(x)=\alpha f(x)+3$ for all $x \in R$,then $f(-\log _e 2)$ is equal to . . . . . . . . .