Find the general solution of the differential equation: $\frac{dy}{dx} - \frac{y}{x} = x^2$

Let $y = y(x)$ be the solution of the differential equation $\frac{dy}{dx} + 2y = f(x)$,where $f(x) = \begin{cases} 1, & x \in [0, 1] \\ 0, & \text{otherwise} \end{cases}$. If $y(0) = 0$,then $y\left(\frac{3}{2}\right)$ is

The solution of the differential equation $(x + \log y)dy + y\,dx = 0$ is:

If a curve passes through the origin and the slope of the tangent to it at any point $(x, y)$ is $\frac{x^{2}-4x+y+8}{x-2}$,then this curve also passes through the point

The integrating factor of the linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$ is a solution of the differential equation:

Let the slope of the tangent line to a curve at any point $P(x, y)$ be given by $\frac{xy^2 + y}{x}$. If the curve intersects the line $x + 2y = 4$ at $x = -2$,then the value of $y$,for which the point $(3, y)$ lies on the curve,is ..... .