In this case, they are not mistaken.
Suppose that the constant is 1-> C[1]=1 then:
sol = DSolve[2 x y[x] - y[x]^2/y'[x] - x^2 y'[x] == 4, y[x], x] /.
C[1] -> 1
Output:-> {{y[x] -> 4 (1 - x)}, {y[x] -> -4 (1 + x)}}
Let's check.The first solution:
Simplify[2 x sol[[1, 1, 2]] - sol[[1, 1, 2]]^2/D[sol[[1, 1, 2]], x] -
x^2 D[sol[[1, 1, 2]], x]] == 4
Output:-> True
The second solution:
Simplify[2 x sol[[2, 1, 2]] - sol[[2, 1, 2]]^2/D[sol[[2, 1, 2]], x] -
x^2 D[sol[[2, 1, 2]], x]] == 4
Output:-> True