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" 4."x^(2)(dy)/(dx)=2xy+y^(2)...

" 4."x^(2)(dy)/(dx)=2xy+y^(2)

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Solve -x^(2)(dy)/(dx)=2xy+y^(2)

x^(2)(dy)/(dx)=x^(2)+xy+y^(2)

x^(2)(dy)/(dx)+y^(2)=xy

(dy)/(dx)=xy+2y+x+2

2xy(dy)/(dx)=x^(2)+y^(2)

The first integral of (dy)/(dx)((d^(2)y)/(dx^(2)))-x^(2)y((dy)/(dx))=xy^(2) will be

(dy)/(dx)=(x^2y+y)/(xy^2+x)