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

Let $y=y(x)$ be the solution of the differential equation $(x^2+1) y^{\prime}-2 x y=(x^4+2 x^2+1) \cos x$,with $y(0)=1$. Then $\int_{-3}^3 y(x) d x$ 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 $S$?

The solution of $(x+y+1) \frac{dy}{dx} = 1$ 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:

Let $f(x)$ be differentiable on the interval $(0, \infty)$ such that $f(1)=1$,and $\lim _{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1$ for each $x>0$. Then $f(x)$ is