A plotting program draws a curve in a plane with plain numbers x and y on its axes…

Step 1. Turn it over. $\dfrac{dx}{dy} = \dfrac{x - x^2y}{y} = \dfrac{x}{y} - x^2$.

Step 2. Substitute. With $v = \dfrac{1}{x}$ we get $\dfrac{dv}{dy} = -\dfrac{1}{x^2}\dfrac{dx}{dy} = -\dfrac{v}{y} + 1$. So $\dfrac{dv}{dy} + \dfrac{v}{y} = 1$.

Step 3. Integrating factor. The factor is $y$, so $\dfrac{d}{dy}(yv) = y$ and $yv = \dfrac{y^2}{2} + C$. That is $\dfrac{y}{x} = \dfrac{y^2}{2} + C$.

Step 4. The constant. At $(1, 1)$: $1 = \dfrac{1}{2} + C$, so $C = 0.5$.

Step 5. Evaluate. Then $x = \dfrac{2y}{y^2 + 1}$. At $y = 3$, $x = \dfrac{6}{10} = 0.6$.

Step 6. The largest $x$. Since $y^2 + 1 \ge 2y$, we get $x \le 1$, with equality at $y = 1$. The largest value is $1$.