3.3.23 Chini first order ode \(y^{\prime }=f\left ( x\right ) \left ( y^{\prime }\right ) ^{n}+g\left ( x\right ) y+h\left ( x\right ) \)

ode internal name "first_order_ode_chini"

This ode is normally generated when we get an Abel ode of first kind \(f_{0}+f_{1}y+f_{2}y^{2}+f_{3}y^{3}\) and then remove the square term \(f_{2}\) using the transformation \(y=u\left ( x\right ) -\frac {f_{2}}{3f_{3}}\). Again as mentioned above, this is done when the Abel invariant is constant. See above section.

Now we check if the Chini invariant is also constant or not. The Chini invariant is given by

\[ \Delta =f^{-n-1}h^{-2n+1}\left ( fh^{\prime }-f^{\prime }h-ngfh\right ) ^{n}n^{-n}\]

And if this comes out to be constant (i.e. do not depend on \(x\)), then we can now solve the Chini ode using method given in Kamke page 303.

Otherwise there is no general method to solve it.  This below is my translation of Kamke 1.55, page 303 on Chini ode. He says, given ode

\begin{equation} y^{\prime }=f\left ( x\right ) \left ( y^{\prime }\right ) ^{n}+g\left ( x\right ) y+h\left ( x\right ) \tag {1}\end{equation}

If for a suitable constants  \(\alpha ,\beta \)

\begin{equation} \left ( \frac {h}{f}\right ) ^{\frac {1}{n}}=e^{\int gdx}\left ( \beta +\alpha \int he^{-\int gdx}dx\right ) \tag {2}\end{equation}

if and when

\begin{equation} z=\left ( \frac {h}{f}\right ) ^{\frac {1}{n}} \tag {3}\end{equation}

A solution of the linear equation

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

you get the solutions of the original ode

\begin{equation} y=\left ( \frac {h}{f}\right ) ^{\frac {1}{n}}u\left ( x\right ) \tag {5}\end{equation}

Through which

\begin{equation} \int \frac {du}{u^{n}-\alpha u+1}+c_{1}=\int \left ( \frac {h}{f}\right ) ^{\frac {1}{n}}hdx \tag {6}\end{equation}

Is determined. For \(h=0\) the ode is Bernoulli. Lets try to figure how the above works on number of examples.