Let $f:[0, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f(x) = 1 - 2x + \int_0^x e^{x-t} f(t) dt$ for all $x \in [0, \infty)$. Then the area of the region bounded by $y = f(x)$ and the coordinate axes is

For any real numbers $\alpha$ and $\beta$,let $y_{\alpha, \beta}(x), x \in R$,be the solution of the differential equation $\frac{dy}{dx}+\alpha y=x e^{\beta x}, y(1)=1$. Let $S=\{y_{\alpha, \beta}(x): \alpha, \beta \in R\}$. Then which of the following functions belong$(s)$ to the set $S$?

For the differential equation $y^2 dx + \left( x - \frac{1}{y} \right) dy = 0$ with the initial condition $y(1) = 1$,find $x$.

For all $x > 0$,let $y_1(x), y_2(x)$,and $y_3(x)$ be the functions satisfying $\frac{dy_1}{dx} - (\sin x)^2 y_1 = 0, y_1(1) = 5$; $\frac{dy_2}{dx} - (\cos x)^2 y_2 = 0, y_2(1) = \frac{1}{3}$; and $\frac{dy_3}{dx} - \left(\frac{2-x^3}{x^3}\right) y_3 = 0, y_3(1) = \frac{3}{5e}$ respectively. Then $\lim_{x \rightarrow 0^{+}} \frac{y_1(x) y_2(x) y_3(x) + 2x}{e^{3x} \sin x}$ is equal to:

Find the solution of the differential equation given below:
$\frac{dy}{dx} + y \cdot \csc^2 (x) = \csc^2 (x) \cdot \cot (x)$

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 . . . . . . . . .