Question

(x²-xy+y²)dx-xydy=0 answer with solution

Answer 1

(x^2 -xy +y^2)dx -xydy =0 
divide by x^2 

(1-y/x +y^2/x^2) dx - y/x dy = 0 
Let y/x = v 
y=vx 

(1-v+v^2) dx - v dy = 0 
dy = (vdx+xdv) 
(1-v+v^2) dx - v (v dx+x dv) = 0 
(1-v+v^2-v^2) dx - vx dv = 0 
(1-v) dx - vx dv = 0 
(1-v) dx = vx dv 

divide both sides by (1-v) x 
dx/x = v/(1-v) dv 

Integrate both sides 
v/(1-v) = 1/(1-v) - 1 

∫ dx/x = ∫ dv/(1-v) - ∫ dv 

Substitute v=y/x 

ln(x) + C1 = -ln(1-v) - v 
ln(x) + C1 = - ln[(x-y)/x] - (y/x) 
ln[(x-y)/x] + (y/x) = - ln(x) + C