Example 2

\[ y^{\prime }=y^{4}+x^{\left ( -\frac {4}{3}\right ) }\]

This one, both Maple and Mathematica can solve. Lets see how. First we check the Chini invariant. It should come out as constant. We see that \(f=1,g=0,h=x^{\left ( -\frac {4}{3}\right ) },n=4\), hence

\begin{align*} \Delta & =f^{-n-1}h^{-2n+1}\left ( fh^{\prime }-f^{\prime }h-ngfh\right ) ^{n}n^{-n}\\ & =1\left ( x^{\left ( -\frac {4}{3}\right ) }\right ) ^{-2\left ( 4\right ) +1}\left ( \frac {d}{dx}\left ( x^{\left ( -\frac {4}{3}\right ) }\right ) -0-0\right ) ^{4}4^{-4}\\ & =\left ( x^{-\frac {4}{3}}\right ) ^{-7}\left ( -\frac {4}{3x^{\frac {7}{3}}}\right ) ^{4}4^{-4}\\ & =4^{-4}x^{\frac {28}{3}}\left ( \frac {4^{4}}{3^{4}}x^{-\frac {28}{3}}\right ) \\ & =4^{-4}\left ( \frac {4^{4}}{3^{4}}\right ) \\ & =\frac {1}{81}\end{align*}

The above \(\Delta \) is also used in the solution below. So we need to find it each time. It is a constant in this example, this is why Maple and Mathematica were able to solve it. Now we follow Kamke method to actually solve the ode. The first thing is to find \(\alpha ,\beta \) such that (2) is true. We see that \(f=1,g=0,h=x^{\left ( -\frac {4}{3}\right ) },n=4\). Now we need to find \(\alpha \). This can be found more easily from EQ (4)

\begin{equation} z^{\prime }-gz=\alpha h \tag {4}\end{equation}

Where \(z=\left ( \frac {h}{f}\right ) ^{\frac {1}{n}}=\left ( \frac {x^{\left ( -\frac {4}{3}\right ) }}{1}\right ) ^{\frac {1}{4}}=x^{-\frac {1}{3}}\). Hence \(z^{\prime }=-\frac {1}{3}x^{-\frac {4}{3}}\). Therefore (4) becomes (given that \(g=0\))

\begin{align*} -\frac {1}{3}x^{-\frac {4}{3}} & =\alpha x^{\left ( -\frac {4}{3}\right ) }\\ \alpha & =-\frac {1}{3}\end{align*}

Since \(\Delta \) is not zero, then solution is directly given as (from Kamke)

\begin{align*} \int ^{\alpha \left ( \frac {h}{f}\right ) ^{\frac {-1}{n}}y\left ( x\right ) }\frac {1}{\frac {u^{n}}{\Delta }-u+1}du-\int \alpha \left ( \frac {h}{f}\right ) ^{\frac {-1}{n}}hdx+c_{1} & =0\\ \int ^{-\frac {1}{3}x^{\frac {1}{3}}y\left ( x\right ) }\frac {1}{81u^{4}-u+1}du+\frac {1}{3}\int x^{\left ( \frac {1}{3}\right ) }x^{\left ( -\frac {4}{3}\right ) }dx+c_{1} & =0\\ \int ^{-\frac {1}{3}x^{\frac {1}{3}}y\left ( x\right ) }\frac {1}{81u^{4}-u+1}du+\frac {1}{3}\int \frac {1}{x}dx+c_{1} & =0\\ \int ^{-\frac {1}{3}x^{\frac {1}{3}}y\left ( x\right ) }\frac {1}{81u^{4}-u+1}du+\frac {1}{3}\ln \left ( x\right ) +c_{1} & =0 \end{align*}