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

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

Integrating factor of the differential equation $\frac{dy}{dx} + y = \frac{1+y}{x}$ is

Let $f:[2,5] \rightarrow R$ be a differentiable function and $\frac{f(5)}{f(2)}=1$. If there is a $c \in (2,5)$ such that $c f^{\prime}(c)=2 f(c)-2 c^3$, then $f(x)=$

The solution of the differential equation $x^2 \frac{dy}{dx} - xy = 1$ is...

If $x = x(y)$ is the solution of the differential equation $y \frac{dx}{dy} = 2x + y^{3}(y+1)e^{y}$ with the initial condition $x(1) = 0$,then $x(e)$ is equal to: