Answer to Question #337828 in Differential Equations for rere

Question

integrating factor by inspection (2x^2 + y)dx + (x^2y - x)dy = 0

Answer 1

integrating factor by inspection:

$\left(2x^2+y\right)dx+\left(x^2y-x\right)dy=0$

Solution:

$\left(2x^2dx+x^2ydy\right)+\left(ydx-xdy\right)=0$,

$\frac{x^2\left(2dx+ydy\right)}{x^2}-\frac{\left(xdy-ydx\right)}{x^2}=0$ ,

$\left(2dx+ydy\right)-\frac{\left(xdy-ydx\right)}{x^2}=0$ ,

$d\left(2x+\frac{y^2}{2}\right)-d\left(\frac{y}{x}\right)=0$ ,

$d\left(2x+\frac{y^2}{2}-\frac{y}{x}\right)=0$ ,

$2x+\frac{y^2}{2}-\frac{y}{x}=C$ ,

Answer: $U\left(x,y\right)=2x+\frac{y^2}{2}-\frac{y}{x}=C$ , integrating factor is $\mu\left(x\right)=\frac{1}{x^2}$