The homogeneous differential equation of the form $\left(1+e^{\frac{x}{y}}\right) dx + e^{\frac{x}{y}}\left(1-\frac{x}{y}\right) dy = 0$ can be solved by making the substitution:

One card is drawn at random from a well-shuffled deck of $52$ cards. In which of the following cases are the events $E$ and $F$ independent?
$E:$ 'The card drawn is a king or a queen'
$F:$ 'The card drawn is a queen or a jack'

The general solution of the differential equation $(3xy+y^2) dx + (x^2+xy) dy = 0$ is

The solution of the differential equation $\frac{dy}{dx} = \frac{x - y}{x + y}$, with the initial condition $y(0) = 0$, represents which of the following curves?

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

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