Find the particular solution of the differential equation $2y e^{x/y}d x+(y-2x e^{x/y})d y=0$ given that x = 0 when y = 1.

Find the solution of the differential equation $(x d y-y d x)\,y\,\mathrm{sin}{\left({\frac{y}{x}}\right)}=(y d x+x d y)x\cos\biggl({\frac{y}{x}}\biggr).$

Solve the following differential equation : $\left(1+e^{\frac{x}{y}}\right)d x+e^{\frac{x}{y}}\left(1-{\frac{x}{y}}\right)d y=0.$

Solve the differential equation : $y+x{\frac{d y}{d x}}=x-y{\frac{d y}{d x}}.$

Solve the following differential equation : ${\frac{d y}{d x}}={\frac{y}{x}}+{\frac{\sqrt{x^{2}+y^{2}}}{x}},\,x>0\,.$

Find the general solution of the differential equation : $(x-y){\frac{d y}{d x}}\,=x+2y.$