An integrating factor of the differential equation $(x^2+1) \frac{dy}{dx} + xy = x^3$ is

The general solution of the differential equation $\frac{dy}{dx} + \frac{y}{x} = 3x$ is

Let $f: R \rightarrow R$ be a continuous function satisfying $f(x)=x+\int_0^x f(t) dt$,for all $x \in R$. Then,the number of elements in the set $S=\{x \in R: f(x)=0\}$ is

Let $f$ be a differentiable function such that $2(x+2)^2 f(x) - 3(x+2)^2 = 10 \int_0^x (t+2) f(t) dt$ for $x \geq 0$. Then $f(2)$ is equal to . . . . . .

Let $y(x)$ be a solution of the differential equation $(1+e^x) y^{\prime}+y e^x=1$. If $y(0)=2$,then which of the following statements is (are) true?
$(A)$ $y(-4)=0$
$(B)$ $y(-2)=0$
$(C)$ $y(x)$ has a critical point in the interval $(-1,0)$
$(D)$ $y(x)$ has no critical point in the interval $(-1,0)$

Suppose $y=y(x)$ is the solution curve to the differential equation $\frac{dy}{dx}-y=2-e^{-x}$ such that $\lim_{x \rightarrow \infty} y(x)$ is finite. If $a$ and $b$ are respectively the $x$- and $y$-intercepts of the tangent to the curve at $x=0$,then the value of $a-4b$ is equal to: