The general solution of the differential equation $\frac{dx}{dt} = \frac{x \log x}{t}$ is

The general solution of the differential equation $\frac{dy}{dx} = e^{x+y} + x^2 e^{x^3+y}$ is (where $C$ is a constant of integration):

The solution of the differential equation $\frac{dy}{dx} = \frac{x-y+3}{2(x-y)+5}$ is

The general solution of the differential equation $\frac{dy}{dx} = \frac{2x-3y+5}{6x-9y+7}$ is

Let a curve $y=f(x)$ pass through the point $(2, (\ln 2)^2)$ and have slope $\frac{2y}{x \ln x}$ for all positive real values of $x$. Then the value of $f(e)$ is equal to:

The solution of the differential equation $(1+e^{-x})(1+y^2) \frac{dy}{dx} = y^2$ which passes through the point $(0,1)$ is