5.`" " x^2dy/dx=2xy+y^2`

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

(xy - x^2)dy = y^2 dx

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

x^2(dy/dx)^2-2xy dy/dx+2y^2-x^2=0

The solution x^2( dy )/( dx) = x^2 + xy + y^2 is :

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

The solution of (dy)/(dx)=2xy-3y+2x-3 is

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