How can I solve this differental equation?(1x−y21(x−y)2)dx+(x21(x−y)2−1y)dy=0&nbsp;?I have this as a part of my homework, and there were like 20 other differential equations that I easily solved, but this one stood out.This is not linear, not separable. I don't know how to approach this problem. If you could tell me to what class this kind of equation belongs, and some methods to solve them, I would be very very glad.

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How can I solve this differental equation?(1x−y21(x−y)2)dx+(x21(x−y)2−1y)dy=0&amp;nbsp;?I have this as a part of my homework, and there were like 20 other differential equations that I easily solved, but this one stood out.This is not linear, not separable. I don&#039;t know how to approach this problem. If you could tell me to what class this kind of equation belongs, and some methods to solve them, I would be very very glad.

Alannah Campos · Accepted Answer

Step 1The equation can be rewritten asdxx-y2(x-y)2dx+x2(x-y)2dy-dyy=0.Step 2Here, I would suggest the substitution&amp;nbsp;xw=y. Hencedxx−w2(1−w)2dx+1(1−w)2d(xw)−d(xw)xw=dxx−w2(1−w)2dx+x(1−w)2dw+w(1−w)2dx−dww−dxx=−w2(1−w)2dx+w(1−w)2dx+x(1−w)2dw−dww=w(1−w)(1−w)2dx+xd(11−w)−d[ln⁡(w)]=w1−wdx+xd(11−w)−d[ln⁡(w)]=−dx+dx1−w+xd(11−w)−d[ln⁡(w)]=d(x1−w)−dx−d[ln⁡(w)]=d[x1−w−x−ln⁡(w)]=d[x−(x−xw)1−w−ln⁡(w)]=d[xw1−w−ln⁡(w)]=d[x2wx−xw−ln⁡(w)]=d[xyx−y−ln⁡(y)+ln⁡(x)]=0

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