Integrating factor of $x \frac{dy}{dx} - y = x^4 - 3x$ is

Let $f(x)$ be a continuously differentiable function on the interval $(0, \infty)$ such that $f(1)=2$ and $\lim _{t \rightarrow x} \frac{t^{10} f(x)-x^{10} f(t)}{t^9-x^9}=1$ for each $x>0$. Then,for all $x>0, f(x)$ is equal to

If $y = y(x)$ is the solution of the differential equation $\frac{dy}{dx} + (\tan x)y = \sin x, 0 \leq x \leq \frac{\pi}{3},$ with $y(0) = 0,$ then $y\left(\frac{\pi}{4}\right)$ is equal to:

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

If $x=x(t)$ is the solution of the differential equation $(t+1) dx = (2x + (t+1)^4) dt$ with the initial condition $x(0) = 2$,then $x(1)$ equals:

Let $y=y(x)$ be the solution of the differential equation $x^3 dy + (xy - 1) dx = 0, x > 0$,with $y(\frac{1}{2}) = 3 - e$. Then $y(1)$ is equal to