Question

Find the integrating factor of given differential equation

(x²y + y²) dx + (y³-x³) dy = 0

Answer 1

This Given Differential Equation is of the form:

$M(x,y)dx+N(x,y)dy=0$

where $M(x,y)=x^2y+y^2$ and $N(x,y)=y^3−x^3$ Now

$Mx−Ny=(x^2y+y^2)x-(y^3-x^3)y$

$\\=x^3y+xy^2-y^4+x^3y\\=2x^3y+xy^2-y^4\\≠0$

Therefore the integrating factor $( I.F.)=\dfrac{1}{Mx-Ny}$

$=\dfrac{1}{2x^3y+xy^2-y^4}$