This is a simulation of doing it by hand step-by-step with your substitution:

(y'[x] == (x^2 + y[x]^2)/(x*y[x] - x^2)) /. y -> Function[t, t*v[t]]
Solve[%, v'[x]][[1, 1]] /. Rule -> Equal // Simplify
DivideSides[%, %[[2]] x, GenerateConditions -> False]
ApplySides[Integrate[#, x] &, %]
SubtractSides[%, 0 == C[1]]
% /. v -> Function[t, y[t]/t]
Solve[%, y[x]]

Mathematica give you solution with explicit form it's a general solving diff equation.

EQ = y'[x] == (x^2 + y[x]^2)/(x*y[x] - x^2);
sol = DSolve[EQ, y, x] // Quiet
EQ /. sol[[1]] // FullSimplify(*A correct solution*)

Your solution by hand are implicit form:

Solve[y == x*Log[((x + y)^2)/(x)] + C[1]*x, y] // PowerExpand

(*{{y -> -x - 
    2 x ProductLog[-(E^(1/2 (-1 - C[1]))/(2 Sqrt[x]))]}, {y -> -x - 
    2 x ProductLog[E^(1/2 (-1 - C[1]))/(2 Sqrt[x])]}}*)

Regards M.I.