Let $f :(0, \infty) \rightarrow R$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f^{\prime}(x)=2 x f(x)+3$,with $f(1)=4$. Then $2 f(2)$ is equal to:

Let the solution $y=y(x)$ of the differential equation $\frac{dy}{dx}-y=1+4 \sin x$ satisfy $y(\pi)=1$. Then $y\left(\frac{\pi}{2}\right)+10$ is equal to:

If $y(x)$ satisfies the differential equation $y' + y = 2(\sin x + \cos x)$ and $y(0) = 1$,then

If $f(x)$ is a differentiable function in the interval $(0, \infty)$ such that $f(1) = 1$ and $\mathop {\lim }\limits_{t \to x} \frac{{{t^2}f(x) - {x^2}f(t)}}{{t - x}} = 1$ for each $x > 0$,then $f(\frac{3}{2})$ is equal to

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

$y+x^2=\frac{dy}{dx}$ has the solution