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

Find the general solution of the differential equation: $\frac{dy}{dx} + 3y = e^{-2x}$

Let $y=y(x)$ be the solution of the differential equation $e^{y} \frac{d y}{d x}-2 e^{y} \sin x+\sin x \cos ^{2} x=0$,with $y(\frac{\pi}{2})=0$. If $y(0)=\log _{e}(\alpha+\beta e^{-2})$,then $4(\alpha+\beta)$ is equal to $....$

By multiplying with $e^{\int P dx}$ on both sides of the equation $\frac{dy}{dx} + P(x)y = Q(x)$,the left side of the equation takes the form $\frac{d}{dx}(y f(x))$,then $f(x) =$

The general solution of the differential equation $(1+y^2) dx = (\tan^{-1} y - x) dy$ is

The solution of the differential equation $2x \frac{dy}{dx} - y = 3$ represents a family of