$f(x, y) = \frac{1}{x + y}$ is a homogeneous function of degree

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 $\frac{y}{x}-\cos ^2 \frac{y}{x}$. Find the equation of the curve.

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

For the differential equation,the general solution for $x \cos \left( \frac{y}{x} \right) (y dx + x dy) = y \sin \left( \frac{y}{x} \right) (x dy - y dx)$,(where $c$ is the constant of integration) is

If $\sin \left(\frac{y}{x}\right)=\log |x|+\frac{\alpha}{2}$ is the solution of the differential equation $x \cos \left(\frac{y}{x}\right) \frac{d y}{d x}=y \cos \left(\frac{y}{x}\right)+x$ and $y(1)=\frac{\pi}{3}$,then $\alpha^2$ is equal to