For (2x2 + 5xy2)dx + (5xy2 − 2y4)dy = 0 or dy/dx = (2x2 + 5xy2)/(2y4−5xy2), consider Homogeneity, Exactness, and the possibility of an Integrating Factor.
The equation dy/dx = (2x2+5xy2)/(2y4−5xy2) is homogeneous if (2(tx)2 + 5(tx)(ty)2)/(2(ty)4 − 5(tx)(ty)2) simplifes to (2x+5xy2)/(2y4−5xy2) for every real number t in some non-empty interval.
Here (2(tx)2 + 5(tx)(ty)2)/(2(ty)4 − 5(tx)(ty)2) reduces to [t2(2x2 + 5txy2)]/[t3(2ty4−5xy2)] or
[(2x2+5txy2)]/[t(2ty4−5xy2)]; t cannot be completely eliminated from [(2x2 + 5txy2)]/[t(2ty4−5xy2)] so
dy/dx = (2x2 + 5xy2)/(2y4−5xy2) is not homogeneous.
The equation (2x2 + 5xy2)dx + (5xy2 − 2y4)dy = 0 is exact if ∂(2x2 + 5xy2)/∂y = ∂(5xy2 − 2y4)/∂x.
Since 10xy is not identical to 5y2, the equation is not exact.
Next consider (10xy − 5y2)/(5xy2 − 2y4) and (10xy − 5y2)/(2x2 + 5xy2), which will not, respectively, reduce to functions of x and y alone. There is then no integrating factor which will multiply through
(2x2 + 5xy2)dx + (5xy2 − 2y4)dy = 0 and give a resulting equation that is exact.
Miscellaneous substitutions (y=vx, y=v/x, x2=u & y2=v, et cetera) can be explored that might reduce
(2x2 + 5xy2)dx + (5xy2 − 2y4)dy = 0 to a rewritten equation with variables that can be separated. However, several available online applications that treat Differential Equations have all failed to return analyses or results for (2x2 + 5xy2)dx + (5xy2 − 2y4)dy = 0.
