The integrating factor of the differential equation $\frac{dy}{dx} + \frac{1}{x}y = x^3 - 3$ is

Let $y=y(x)$ be a differentiable function in the interval $(0, \infty)$ such that $y(1)=2$ and $\lim_{t \rightarrow x} \left( \frac{t^{2}y(x)-x^{2}y(t)}{x-t} \right) = 3$ for each $x>0$. Then $2y(2)$ is equal to

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:

An integrating factor for the differential equation $(1 + y^2)dx - (\tan^{-1} y - x)dy = 0$ is:

Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx} = \frac{y}{x}(1 + xy^2(1 + \log_e x))$ for $x > 0$ and $y(1) = 3$. Then $\frac{y^2(x)}{9}$ is equal to:

The general solution of the differential equation $(y^2+x+1) dy = (y+1) dx$ is