The Integrating Factor of the differential equation $x \frac{dy}{dx} + 2y = x^2$ $(x \neq 0)$ is . . . . . . .

If the form of the solution of the differential equation $(y^3+x) \frac{dy}{dx} = y$ with the condition $y(4) = 2$ is $y^3 = ax + b$,then $4a + 12b^2 = $

Find a particular solution of the differential equation $\frac{dy}{dx} + y \cot x = 4x \csc x$ $(x \neq 0),$ given that $y=0$ when $x=\frac{\pi}{2}$.

The general solution of the differential equation $\frac{dy}{dx} + \left(\frac{3x^2}{1+x^3}\right)y = \frac{1}{x^3+1}$ is

Let us consider a curve $y=f(x)$ passing through the point $(-2, 2)$ and the slope of the tangent to the curve at any point $(x, f(x))$ is given by $f(x)+x f'(x)=x^2$. Then:

If $y(t)$ is a solution of $(1 + t)\frac{dy}{dt} - ty = 1$ and $y(0) = -1$,then $y(1)$ is equal to