If $y=f(x)$ is the solution of the differential equation $x \frac{dy}{dx} = x^2 + 3y$, $x > 0$, $y(2) = 4$, then $f(4) = $ ?

Let $f$ be a non-negative function defined on $\left[0, \frac{\pi}{2}\right]$. If $\int_0^x \left(f^{\prime}(t)-\sin 2t\right) dt = \int_x^0 f(t) \tan t dt$ and $f(0)=1$, then find $\int_0^{\frac{\pi}{2}} f(x) dx$.

Find the general solution of the differential equation $x \frac{dy}{dx} + 2y = x^2$ where $x \neq 0$.

If $y=y(x)$ is the solution curve of the differential equation $\frac{dy}{dx} + y \tan x = x \sec x$,$0 \leq x \leq \frac{\pi}{3}$,with $y(0)=1$,then $y\left(\frac{\pi}{6}\right)$ is equal to

The solution of the equation $\frac{dy}{dx} + 2y \tan x = \sin x$ satisfying $y = 0$ when $x = \frac{\pi}{3}$ is:

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