Integration of Total Differentials

Integrate the equation $(x+y)\, dx+(x-y)\, dy=0$.

Integrate the equation $(3x^{2}+6xy^{2})\, dx+(6x^{2}y+4y^{3})\, dy=0$.

Integrate the equation $(x^{3}-3xy^{2})\, dx+(y^{3}-3x^{2}y)\, dy=0$.

Integrate the equation $\dfrac {x\, dx+y\, dy}{\sqrt{x^{2}+y^{2}}}+\dfrac {x\, dy-y\, dx}{x^{2}}=0$.

Integrate the equation $x\, dx+y\, dy+\dfrac {x\, dy-y\, dx}{x^{2}+y^{2}}=0$.

Having found the general integral of the equation $\dfrac {2x\, dx}{y^{3}}+\dfrac {y^{2}-3x^{2}}{y^{4}}\, dy=0$, choose the constant $C$ in it so as to obtain the curve passing through the point $M(1,1)$.

Having found the general integral of the equation $\left(1+e^{\frac{x}{y}}\right)dx+e^{\frac{x}{y}}\left(1-\frac{x}{y}\right)dy=0$, choose the constant $C$ in it so as to obtain the curve passing through the point $M(1,1)$.