1. **State the problem:** Find the general solution of the differential equation given implicitly by $$y = px + \frac{a}{p}$$ where $p = \frac{dy}{dx}$. 2. **Rewrite the equation:** Substitute $p = \frac{dy}{dx}$ into the equation: $$y = x \frac{dy}{dx} + \frac{a}{\frac{dy}{dx}} = x p + \frac{a}{p}$$ 3. **Multiply both sides by $p$ to clear the denominator:** $$y p = x p^2 + a$$ 4. **Rearrange to isolate terms:** $$x p^2 - y p + a = 0$$ 5. **This is a quadratic equation in $p$:** $$x p^2 - y p + a = 0$$ 6. **Solve for $p$ using the quadratic formula:** $$p = \frac{y \pm \sqrt{y^2 - 4 a x}}{2 x}$$ 7. **Recall that $p = \frac{dy}{dx}$, so:** $$\frac{dy}{dx} = \frac{y \pm \sqrt{y^2 - 4 a x}}{2 x}$$ 8. **Separate variables or rewrite to solve the differential equation:** This implicit form represents the general solution of the differential equation. **Final answer:** The general solution satisfies $$x \left(\frac{dy}{dx}\right)^2 - y \frac{dy}{dx} + a = 0$$ or equivalently $$\frac{dy}{dx} = \frac{y \pm \sqrt{y^2 - 4 a x}}{2 x}$$