solve the differential equation (x^2-xy+y^2)dy-xydx=0
plz show me working
5 answers
The last time I did differential equations was 55 years ago, perhaps one of our other math tutors can kick in here
http://www.wolframalpha.com/input/?i=solve+x+y+dx+%3D+(x%5E2+-x+y+%2By%5E2)dy
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But, note that
y' = xy/(x^2-xy+y^2)
This is homogeneous, so let y=vx
Then y' = v + xv'
v + xv' = vx^2/(x^2-vx+x^2v^2)
v + xv' = v/(v^2-v+1)
xv' = (v^2-2v+1)/(v^2-v+1)
(v^2-v+1)/(v-1)^2 dv = -1/x dx
Now just do a long division and partial fractions, and integrate both sides. Then substitute back using v = y/x.
But, note that
y' = xy/(x^2-xy+y^2)
This is homogeneous, so let y=vx
Then y' = v + xv'
v + xv' = vx^2/(x^2-vx+x^2v^2)
v + xv' = v/(v^2-v+1)
xv' = (v^2-2v+1)/(v^2-v+1)
(v^2-v+1)/(v-1)^2 dv = -1/x dx
Now just do a long division and partial fractions, and integrate both sides. Then substitute back using v = y/x.