The general solution of the differential equation $y(1+\log x)\left(\frac{dx}{dy}\right) - x \log x = 0$ is

The slope of a curve at any point is inversely proportional to twice the $y$-coordinate of that point. If the curve passes through $(4, 3)$,then the equation of the curve is:

The solution of ${e^{dy/dx}} = (x + 1)$,$y(0) = 3$ 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:

Let $y=y(x)$ be the solution of the differential equation $\log _{e}\left(\frac{d y}{d x}\right)=3 x+4 y$,with $y(0)=0$. If $y\left(-\frac{2}{3} \log _{e} 2\right)=\alpha \log _{e} 2$,then the value of $\alpha$ is equal to:

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