The Integrating Factor

Find the integral of the equation $(x^{2}+y)\, dx-x\, dy=0$, which has an integrating factor of one of the forms $M=f(x)$ or $M=f(y)$.

Find the integral of the equation $(x^{2}+y^{2})(x\, dy-y\, dx)=(a+x)x^{4}\, dx$, which has an integrating factor of one of the forms $M=f(x)$ or $M=f(y)$.

Find the integral of the equation $(xy^{2}+y)\, dx-x\, dy=0$, which has an integrating factor of one of the forms $M=f(x)$ or $M=f(y)$.

Find the integral of the equation $(2xy^{2}-y)\, dx+(y^{2}+x+y)\, dy=0$, which has an integrating factor of one of the forms $M=f(x)$ or $M=f(y)$.

Find the integral of the equation $(1-x^{2}y)\, dx+x^{2}(y-x)\, dy=0$, which has an integrating factor of one of the forms $M=f(x)$ or $M=f(y)$.

Find the integral of the equation $(x^{2}+y^{2}+2x)\, dx+2y\, dy=0$, which has an integrating factor of one of the forms $M=f(x)$ or $M=f(y)$.

Find the integral of the equation $(x\cos y-y\sin y)\, dy+(x\sin y+y\cos y)\, dx=0$, which has an integrating factor of one of the forms $M=f(x)$ or $M=f(y)$.

Integrate the equation $(x^{2}+x^{2}y+2xy-y^{2}-y^{3})\, dx+(y^{2}+xy^{2}+2xy-x^{2}-x^{3})\, dy=0$ with a factor of one of the forms $M=f(x+y)$ or $M=f(xy)$.

Integrate the equation $(2x^{3}y^{2}-y)\, dx+(2x^{2}y^{3}-x)\, dy=0$ with a factor of one of the forms $M=f(x+y)$ or $M=f(xy)$.

Integrate the equation $xy^{2}\, dx+(x^{2}y-x)\, dy=0$ with a factor of one of the forms $M=f(x+y)$ or $M=f(xy)$.

Integrate the equation $x^{2}y^{3}+y+(x^{3}y^{2}-x)y'=0$ with a factor of one of the forms $M=f(x+y)$ or $M=f(xy)$.

Solve the equation $(x^{2}+y^{2}+1)\, dx-2xy\, dy=0$ using a factor of one of the forms $M=f(x^{2}-y^{2})$ or $M=f(x^{2}+y^{2})$.

Solve the equation $(y+x^{2})\, dy+(x-xy)\, dx=0$ using a factor of one of the forms $M=f(x^{2}-y^{2})$ or $M=f(x^{2}+y^{2})$.

Solve the equation $\omega (x^{2}+y^{2})x\, dx+\omega_{1}(x^{2}+y^{2})y\, dy=0$ using a factor of one of the forms $M=f(x^{2}-y^{2})$ or $M=f(x^{2}+y^{2})$.

Find the integral of the equation $(x^{2}+y^{2}+y)\, dx-x\, dy=0$.

Find the integral of the equation $x\, dy+y\, dx+y^{2}(x\, dy-y\, dx)=0$.

Find the integral of the equation $y(x^{2}+y^{2})\, dx+x(x\, dy-y\, dx)=0$.

Find the integral of the equation $y^{2}(x+a)\, dx+x(x^{2}-ay)\, dy=0$.

Find the integral of the equation $2x\, dy+y\, dx+xy^{2}(x\, dy+y\, dx)=0$.

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

Prove that for the homogeneous equation $M\, dx+N\, dy=0$, the quantity $\dfrac {1}{Mx+Ny}$ is an integrating factor.

Prove the theorem: if the equation $M\, dx+N\, dy=0$ is homogeneous, and $M\, dx+N\, dy$ is a total differential, then the general integral of the equation can be written in the form $Mx+Ny=C$.