The integral curve satisfying $y' = \frac{x^2 + y^2}{x^2 - y^2}$ with $y(1) = 2$ has a slope at the point $(1, 0)$ equal to:

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

Suppose that $f(x, y)$ and $g(x, y)$ are homogeneous functions of the same order. If $x=Vy$ reduces the equation $\frac{dy}{dx}=\frac{f(x, y)}{g(x, y)}$ to the form $\frac{dV}{dy}=\frac{1}{y}(F(V))$,then $F(V)=$

$A$ homogeneous differential equation of the form $\frac{dx}{dy} = h\left(\frac{x}{y}\right)$ can be solved by making the substitution:

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

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