Example 1

\[ y^{\prime }=3y^{4}+x^{3}\]

This one, Maple nor Mathematica can solve. Lets see why. First we check the Chini invariant. We see that \(f=3,g=0,h=x^{3},n=4\), hence

\begin{align*} \Delta & =f^{-n-1}h^{-2n+1}\left ( fh^{\prime }-f^{\prime }h-ngfh\right ) ^{n}n^{-n}\\ & =3^{-4-1}\left ( x^{3}\right ) ^{-2\left ( 4\right ) +1}\left ( 3\left ( 3x^{2}\right ) -0-0\right ) ^{4}4^{-4}\\ & =3^{-5}\left ( x^{3}\right ) ^{-7}\left ( 9x^{2}\right ) ^{4}4^{-4}\\ & =3^{-5}4^{-4}x^{-21}9^{4}x^{8}\\ & =3^{-5}4^{-4}9^{4}x^{-13}\end{align*}

Since Chini invariant then it can’t be solved using Kamke shown method on page 303. To verify, let us try to solve it using Kamke method and see what happens.

The first thing is to find \(\alpha ,\beta \) such that (2) is true. EQ (2) becomes

\begin{align*} \left ( \frac {h}{f}\right ) ^{\frac {1}{n}} & =e^{\int gdx}\left ( \beta +\alpha \int he^{-\int gdx}dx\right ) \\ \left ( \frac {x^{3}}{3}\right ) ^{\frac {1}{4}} & =e^{\int 0dx}\left ( \beta +\alpha \int x^{3}e^{-\int 0dx}dx\right ) \\ & =\beta +\alpha \int x^{3}dx\\ & =\beta +\alpha \frac {x^{4}}{3}\end{align*}

If we set \(\beta =0\) then

\[ \left ( \frac {x^{3}}{3}\right ) ^{\frac {1}{4}}=\alpha \left ( \frac {x^{4}}{3}\right ) \]

We see it is not possible to find constant \(\alpha \) to satisfy this. So we must always check the Chini invariant before trying, this will save time.