2.12.1.6 Example 6 \(\left ( y^{\prime }\right ) ^{2}=4y-x^{2}\)

Solve

\[ \left ( y^{\prime }\right ) ^{2}=4y-x^{2}\]

Hence

\[ y^{\prime }=\pm \sqrt {4y-x^{2}}\]

For the first ode, the first step is to identify if this is class G and find \(F\). We start by multiplying the RHS by \(\frac {x}{y}\) (regardless of what is in the RHS) which gives

\begin{align*} y^{\prime } & =\frac {x}{y}\sqrt {4y-x^{2}}\\ & =F\left ( x,y\right ) \end{align*}

Next we check if \(F\left ( x,y\right ) \) has \(y\) or not in it. If so, then let the RHS above be \(F\left ( x,y\right ) \) and now do

\begin{align*} f_{x} & =x\frac {\partial F}{\partial x}\\ & =-\frac {2x\left ( x^{2}-2y\right ) }{y\sqrt {4y-x^{2}}}\end{align*}

And let

\begin{align*} f_{y} & =y\frac {\partial F}{\partial y}\\ & =\frac {x\left ( x^{2}-2y\right ) }{y\sqrt {4y-x^{2}}}\end{align*}

Now we check, if \(f_{y}=0\) then this is not Homogeneous type G. Else we now need to determine value of \(\alpha \). This is done as follows.

\begin{align*} \alpha & =\frac {fx}{f_{y}}\\ & =-2 \end{align*}

If \(\alpha \) comes out not to have in it \(x\) nor \(y\) as in this case, then we are done. This ode is Homogeneous type G and the ode can be written as

\[ y^{\prime }=\frac {y}{x}F\left ( \frac {y}{x^{\alpha }}\right ) \]

Hence the solution is

\begin{equation} \ln x-c_{1}+\int ^{yx^{\alpha }}\frac {1}{\tau \left ( -\alpha -F\left ( \tau \right ) \right ) }d\tau =0 \tag {1}\end{equation}

Now let \(y=\frac {\tau }{x^{\alpha }}\) and substitute this into \(F\left ( x,y\right ) \) which results in

\begin{align*} F\left ( \tau \right ) & =\frac {x}{y}\sqrt {4y-x^{2}}\\ & =\frac {x}{\frac {\tau }{x^{\alpha }}}\sqrt {4\frac {\tau }{x^{\alpha }}-x^{2}}\\ & =\frac {x}{\tau x^{2}}\sqrt {4\tau x^{2}-x^{2}}\end{align*}

Since the requirement is that \(F\left ( \tau \right ) \) ends up free of \(x\), then the only way to use this method and simplify the above to eliminate \(x\) is to assume \(x>0\). Now the above simplifies to

\[ F\left ( \tau \right ) =\frac {1}{\tau }\sqrt {4\tau -1}\]

The solution(1) becomes

\begin{align*} \ln x-c_{1}+\int ^{yx^{\alpha }}\frac {1}{\tau \left ( -\alpha -F\left ( \alpha \right ) \right ) }d\tau & =0\\ \ln x-c_{1}+\int ^{\frac {y}{x^{2}}}\frac {1}{\tau \left ( 2-\frac {1}{\tau }\sqrt {4\tau -1}\right ) }d\tau & =0\\ \ln x-c_{1}+\int ^{\frac {y}{x^{2}}}\frac {1}{2\tau -\sqrt {4\tau -1}}d\tau & =0 \end{align*}

Solving the integral gives long complicated expression which is verified correct. So better to keep the solution implicit as the above. Now we solve the second ode \(y^{\prime }=-\sqrt {4y-x^{2}}\) in similar way.