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:

The slope of the tangent at any point $(x, y)$ on a curve $y = y(x)$ is $\frac{x^2+y^2}{2xy}$,where $x > 0$. If $y(2) = 0$,then a value of $y(8)$ is

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

Which one of the following functions is not homogeneous?

The general solution of $y^2 dx + (x^2 - xy + y^2) dy = 0$ is:

If the solution curve of the differential equation $\frac{dy}{dx} = \frac{x+y-2}{x-y}$ passing through the point $(2,1)$ is $\tan^{-1}\left(\frac{y-1}{x-1}\right) - \frac{1}{\beta} \log_e\left(\alpha + \left(\frac{y-1}{x-1}\right)^2\right) = \log_e|x-1|$,then $5\beta + \alpha$ is equal to