The particular solution of the differential equation $y(1+\log x) \frac{dx}{dy} - x \log x = 0$ given that $y = e^2$ when $x = e$ is:

The general solution of the differential equation $y y^{\prime} = x \left[ \frac{y^2}{x^2} + \frac{\phi\left(\frac{y^2}{x^2}\right)}{\phi^{\prime}\left(\frac{y^2}{x^2}\right)} \right]$,where $\phi$ is an arbitrary function,is

If a curve passes through the point $\left( 2, \frac{7}{2} \right)$ and has slope $\left( 1 - \frac{1}{x^2} \right)$ at any point $(x, y)$ on it,then the ordinate of the point on the curve whose abscissa is $-2$ is

General solution of $(x+y)^{2} \frac{d y}{d x}=a^{2}, a \neq 0$ is ($C$ is an arbitrary constant)

The solution of $\frac{dy}{dx} = \sqrt{1-y^2}$ with the initial condition $y(0) = 1$ is:

The general solution $y(x)$ of the differential equation $\frac{d}{dy} \left( \int_x^y dt \right) = x$ is: