$\frac{dy}{dx} = \frac{y + x \tan(\frac{y}{x})}{x} \Rightarrow \sin(\frac{y}{x}) = $

If the solution curve $y=y(x)$ of the differential equation $y^{2} dx + (x^{2} - xy + y^{2}) dy = 0$ passes through the point $(1, 1)$ and intersects the line $y = \sqrt{3}x$ at the point $(\alpha, \sqrt{3}\alpha)$,then the value of $\log_{e}(\sqrt{3}\alpha)$ is equal to

If $y=y(x)$ is the solution of $x \frac{d y}{d x}=y+x e^{-\left(\frac{y}{x}\right)}$ with $y(1)=\log e$,then find $y(e)$.

The general solution of the differential equation $\frac{d y}{d x}=\frac{x+7 y+3}{3 x+5 y+9}$ is

The substitution $x=vy$ converts which one of the following differential equations into an equation solvable by the variable separable method?

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: