Example 1 \begin {equation} y^{\prime \prime }+\left ( y^{\prime }\right ) ^{2}+y^{\prime }=0 \tag {1} \end {equation} Let \(p=y^{\prime }\) then \(y^{\prime \prime }=p^{\prime }\). Hence the ode becomes\begin {equation} p^{\prime }+p^{2}+p=0 \tag {2} \end {equation} Which is now a first order separable ode. Its solution can be easily found to be\[ p=\frac {1}{c_{1}e^{x}-1}\] Hence\[ y^{\prime }\left ( x\right ) =\frac {1}{c_{1}e^{x}-1}\] Which is now solved for \(y\left ( x\right ) \) as first order, which gives by integration\[ y=\ln \left ( c_{1}e^{x}-c_{2}+1\right ) -x \]