Example 1 y′′ + (y′)2 + y′ = 0

\begin{equation} y^{\prime \prime }+\left ( y^{\prime }\right ) ^{2}+y^{\prime }=0 \tag {1}\end{equation}

Let \(p=y^{\prime }\) then \(y^{\prime \prime }=\frac {dp}{dx}\). Hence the ode becomes

\begin{equation} \frac {dp}{dx}+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

\[ y=\ln \left ( c_{1}e^{x}-c_{2}+1\right ) -x \]