If the general solution of $\frac{dy}{dx} = \frac{y^2}{xy - y^2 - x^2}$ is $\tan^{-1}\left(\frac{y}{x}\right) = f(y) + C$,then $f(e^3) = $

Let $A$ and $B$ be two independent events. If $P(A) = \frac{1}{5}$ and $P(A \cup B) = \frac{7}{10}$,then what is the value of $P(\overline{B})$?

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{4y^3+2yx^2}{3xy^2+x^3}$ with $y(1)=1$. If for some $n \in N$,$y(2) \in [n-1, n)$,then $n$ is equal to...

The slope of the tangent at $(x, y)$ to a curve passing through $\left( 1, \frac{\pi}{4} \right)$ is given by $\frac{y}{x} - \cos^2\left( \frac{y}{x} \right)$. Then the equation of the curve is:

Show that the differential equation $y^{\prime} = \frac{x+y}{x}$ is a homogeneous equation and find its general solution.

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