The solution of the differential equation $2xy \frac{dy}{dx} = x^2 + 3y^2$ is (where $p$ is a constant):

Show that the differential equation $\left(x^{2}+x y\right) d y=\left(x^{2}+y^{2}\right) d x$ is a homogeneous equation and find its solution.

Let $y=y(x)$ be the solution of the differential equation $x \tan \left(\frac{y}{x}\right) d y=\left(y \tan \left(\frac{y}{x}\right)-x\right) d x$ for $-1 \leq x \leq 1$,with the initial condition $y\left(\frac{1}{2}\right)=\frac{\pi}{6}$. Then the area of the region bounded by the curves $x=0$,$x=\frac{1}{\sqrt{2}}$,and $y=y(x)$ in the upper half plane is:

Show that the differential equation $(x^{2}-y^{2}) dx + 2xy dy = 0$ is a homogeneous equation and find its solution.

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

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.