f[x_, y_] := 2 x y - y^2

sol = DSolve[y'[x] ⩵ f[x, y[x]], y, x] /. C[1] → c

Show[VectorPlot[{1, f[x, y]},

{x, -3, 3}, {y, -3, 4}, VectorStyle → Arrowheads[0.026]],

Plot[Evaluate[Table[y[x] /. sol, {c, -10, 10, 1}]], {x, -2, 2}, PlotRange → All]]

y → Function{x},1

2+

3

22/3 54 - 648 x2 + 648 c + -2916 + 54 - 648 x2 + 648 c21/3

+

54 - 648 x2 + 648 c + -2916 + 54 - 648 x2 + 648 c21/3

6 × 21/3,

y → Function{x},1

2-

3 1 + ⅈ 3

2 × 22/3 54 - 648 x2 + 648 c + -2916 + 54 - 648 x2 + 648 c21/3

-

1 - ⅈ 3 54 - 648 x2 + 648 c + -2916 + 54 - 648 x2 + 648 c21/3

12 × 21/3,

y → Function{x},1

2-

3 1 - ⅈ 3

2 × 22/3 54 - 648 x2 + 648 c + -2916 + 54 - 648 x2 + 648 c21/3

-

1 + ⅈ 3 54 - 648 x2 + 648 c + -2916 + 54 - 648 x2 + 648 c21/3

12 × 21/3

-3 -2 -1 0 1 2 3

-2

0

2

4

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