1.2.2 Example 2 \(y-xy^{\prime }-y^{\prime }+\left ( y^{\prime }\right ) ^{2}=0\)

Given

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

This problem from chapter 7, problem 7. From Boole book, page 137. This is actually a clairaut ode. Let \(y^{\prime }=p\). The above becomes

\begin{equation} y-xp-p+p^{2}=0 \tag {1}\end{equation}

We start by isolating \(x\) which gives

\begin{align*} -xp & =\frac {p-p^{2}}{y}\\ x & =\frac {p-1}{y}\\ & =f\left ( y,p\right ) \end{align*}

Using (3A) and not (2A)

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

This is non-linear ode in \(y\). So this is no better than what we started.

Let try to isolate \(y\) instead. Solving (1) for \(y\) gives

\begin{align*} y & =xp+p-p^{2}\\ & =f\left ( x,p\right ) \end{align*}

Therefore, using (2A) gives

\[ \frac {dx}{dp}=\frac {\frac {df}{dp}}{p-\frac {\partial f}{\partial x}}\]

But \(p-\frac {\partial f}{\partial x}=0\). Hence this method does not work for this ode. The method of Clairaut works on this, since there we apply a different algorithm. It is best to keep the parmateric algorithm separate. So we try this and if this fails, then try other approaches.