Needs["SpecialFunctions`"]

DE1 = AlgebraicDE[(x^2) + (y^2) - (r^2), y[x]]

DE2 = AlgebraicDE[(((y - r))^2) + (x^2) - (r^2), y[x]]

DE3 = AlgebraicDE[((((y^2) - (x^2)))^4) - (x^2)*(y^2), y[x]]

DEtoRE[DE1, y[x], a[k]]

DEtoRE[DE2, y[x], a[k]]

DEtoRE[DE3, y[x], a[k]]