Question #337828

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


1
Expert's answer
2022-05-09T18:21:47-0400

integrating factor by inspection:

(2x2+y)dx+(x2yx)dy=0\left(2x^2+y\right)dx+\left(x^2y-x\right)dy=0

Solution:

(2x2dx+x2ydy)+(ydxxdy)=0\left(2x^2dx+x^2ydy\right)+\left(ydx-xdy\right)=0,

x2(2dx+ydy)x2(xdyydx)x2=0\frac{x^2\left(2dx+ydy\right)}{x^2}-\frac{\left(xdy-ydx\right)}{x^2}=0 ,

(2dx+ydy)(xdyydx)x2=0\left(2dx+ydy\right)-\frac{\left(xdy-ydx\right)}{x^2}=0 ,

d(2x+y22)d(yx)=0d\left(2x+\frac{y^2}{2}\right)-d\left(\frac{y}{x}\right)=0 ,

d(2x+y22yx)=0d\left(2x+\frac{y^2}{2}-\frac{y}{x}\right)=0 ,

2x+y22yx=C2x+\frac{y^2}{2}-\frac{y}{x}=C ,

Answer: U(x,y)=2x+y22yx=CU\left(x,y\right)=2x+\frac{y^2}{2}-\frac{y}{x}=C , integrating factor is μ(x)=1x2\mu\left(x\right)=\frac{1}{x^2}

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