4.9.2.1.6 Example 5 \(y^{\prime \prime }=\left ( y^{\prime }\right ) ^{2}\cos x\)

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

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

\begin{align*} p^{\prime } & =p^{2}\cos x\\ \int \frac {dp}{p^{2}} & =\int \cos xdx\\ -\frac {1}{p} & =\sin x+c_{1}\end{align*}

Hence \(p=\frac {-1}{\sin x+c_{1}}\). But \(p=y^{\prime }\left ( x\right ) \). Therefore

\begin{align*} y^{\prime }\left ( x\right ) & =\frac {-1}{\sin x+c_{1}}\\ \int dy & =-\int \frac {dx}{\sin x+c_{1}}\\ y & =\frac {-\arctan \left ( \frac {2c_{1}\tan \left ( \frac {x}{2}\right ) +2}{2\sqrt {c_{1}^{2}-1}}\right ) }{\sqrt {c_{1}^{2}-1}}+c_{2}\end{align*}