The integrating factor of the differential equation $\frac{dy}{dx} + \frac{y}{x} = x^2$ is . . . . . . .

The solution of the differential equation $\sqrt{1-y^2} dx + x dy - \sin^{-1} y dy = 0$ is

If $y=y(x)$ is the solution of the differential equation $x \frac{dy}{dx} + 2y = x^2$ satisfying $y(1) = 1$,then the value of $y\left(\frac{1}{2}\right)$ is

Let $y^{\prime}(x) + y(x) g^{\prime}(x) = g(x) g^{\prime}(x)$,$y(0) = 0$,$x \in \mathbb{R}$,where $f^{\prime}(x)$ denotes $\frac{d f(x)}{d x}$ and $g(x)$ is a given non-constant differentiable function on $\mathbb{R}$ with $g(0) = g(2) = 0$. Then the value of $y(2)$ is

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:

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