1.2.5 Example 5 \(x=1+y^{\prime }+\left ( y^{\prime }\right ) ^{3}\)

Note: This can also be solved as quadrature, but solving for \(y^{\prime }\) which will generate 3 ode’s, each can be directly integrated. We can only isolate \(x\) here and use (3A) since there is no \(y\).

\begin{align} x & =1+y^{\prime }+\left ( y^{\prime }\right ) ^{3}\nonumber \\ & =1+p+p^{3} \tag {1}\end{align}

Using (3A)

\begin{align*} \frac {dy}{dp} & =\frac {p\frac {df}{dp}}{1-p\frac {\partial f}{\partial y}}\\ & =p\left ( 1+3p^{2}\right ) \end{align*}

This ode is quadrature. Solving gives

\begin{align} y & =\int p\left ( 1+3p^{2}\right ) dp\nonumber \\ & =\frac {p^{2}}{2}+\frac {3}{4}p^{4}+c \tag {2}\end{align}

\(p\) is eliminated between (1,2) to obtain the final solution. This gives implicit solution for \(y\) as

\[ 27x^{4}+64c^{3}-192c^{3}y+72cx^{2}+192cy^{2}-108x^{3}-72x^{2}y-64y^{3}+32c^{2}-144cx-64cy+164x^{2}+144xy+32y^{2}+76c-112x-76y+29=0 \]