Example 2 \begin {equation} y^{\prime \prime }+ay\left ( y^{\prime }\right ) +by^{3}=0 \tag {1} \end {equation} Let \(p=y^{\prime }\) then \(y^{\prime \prime }=p\frac {dp}{dy}\). Hence the ode becomes\begin {equation} p\frac {dp}{dy}+ayp+by^{3}=0 \tag {2} \end {equation} Which is now a first order ode. \begin {equation} \frac {dp}{dy}=-ay+b\frac {y^{3}}{p} \tag {3} \end {equation} Solving for \(p\) gives\[ \frac {1}{4\sqrt {a^{2}+8b}}\left ( \ln \left ( -by^{4}+ay^{2}p+2p^{2}\right ) \sqrt {a^{2}+8b}+2a\operatorname {arctanh}\left ( \frac {ax^{2}+4p}{y^{2}\sqrt {a^{2}+8b}}\right ) \right ) =c_{1}\] Then \(y\) is found by solving \(y^{\prime }=p\), another first order ode.\[ \frac {1}{4\sqrt {a^{2}+8b}}\left ( \ln \left ( -by^{4}+ay^{2}y^{\prime }+2\left ( y^{\prime }\right ) ^{2}\right ) \sqrt {a^{2}+8b}+2a\operatorname {arctanh}\left ( \frac {ax^{2}+4y^{\prime }}{y^{2}\sqrt {a^{2}+8b}}\right ) \right ) =c_{1}\] But this second one could not solve. Actually ode (3) is homogeneous, class G and should use formula given in Kamke’s book, p. 19. but I have yet to implement this.