The solution of the differential equation $x \, dy - y \, dx = 0$ represents a

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

Find the equation of a curve passing through the point $(-2, 3),$ given that the slope of the tangent to the curve at any point $(x, y)$ is $\frac{2x}{y^2}$.

Let a curve $y=f(x)$ pass through the points $(0,5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3+y) e^{2x} dx - (7+e^{2x}) dy = 0$,then $k$ is equal to