By taking $\displaystyle y=px^2+qx+r$, you get the solution $\displaystyle y=x^2+x+r$, where $\displaystyle r\in\mathbb{R}$ is a constant. (Is that right? I hope I didn't mess up the calculations..

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However, $\displaystyle y=x^2+x-a\exp(-x)+b$, where $\displaystyle a,b\in\mathbb{R}$ are constants is also a valid solution, but it doesn't fit in the form that you obtained. So, using that method, you don't arrive to

*all* the solutions. Just thought I'd point that out, I know that's not what you asked..