3.5.5 Special case. \(\left ( y^{\prime }\right ) ^{\frac {n}{m}}=f\left ( x\right ) g\left ( y\right ) \)

ode internal name "first_order_nonlinear_p_but_separable"

For the special case of \(\left ( y^{\prime }\right ) ^{\frac {n}{m}}=F\left ( x,y\right ) \) where RHS is separable, i.e. \(F\left ( x,y\right ) =f\left ( x\right ) g\left ( y\right ) \) then short cut method is described below.  This only works if \(F\left ( x,y\right ) \) is separable and if there is only one \(y^{\prime }\) in the equation. For example, it will not work on \(\left ( y^{\prime }\right ) ^{\frac {3}{2}}+y^{\prime }=yx\) and will not work on \(\left ( y^{\prime }\right ) ^{\frac {3}{2}}=y+x\) (see second special case below for the form \(\left ( y^{\prime }\right ) ^{\frac {n}{m}}=ax+by+c\))

If the form is \(\left ( y^{\prime }\right ) ^{\frac {n}{m}}=f\left ( x\right ) g\left ( y\right ) \) then we first write it as \(\left ( y^{\prime }\right ) ^{n}=\left ( f\left ( x\right ) g\left ( y\right ) \right ) ^{m}\) assuming \(f\left ( x\right ) g\left ( y\right ) >0\). Then find roots on unity for \(n\). For example of \(n=2\) this gives

\[ y^{\prime }=\left \{ \begin {array} [c]{c}\left ( f\left ( x\right ) g\left ( y\right ) \right ) ^{\frac {m}{2}}\\ -\left ( f\left ( x\right ) g\left ( y\right ) \right ) ^{\frac {m}{2}}\end {array} \right . \]

And if \(n=3\) then

\[ y^{\prime }=\left \{ \begin {array} [c]{c}\left ( f\left ( x\right ) g\left ( y\right ) \right ) ^{\frac {m}{3}}\\ -\left ( -1\right ) ^{\frac {1}{3}}\left ( f\left ( x\right ) g\left ( y\right ) \right ) ^{\frac {m}{3}}\\ \left ( -1\right ) ^{\frac {2}{3}}\left ( f\left ( x\right ) g\left ( y\right ) \right ) ^{\frac {m}{3}}\end {array} \right . \]

And if \(n=4\) then

\[ y^{\prime }=\left \{ \begin {array} [c]{c}\left ( f\left ( x\right ) g\left ( y\right ) \right ) ^{\frac {m}{4}}\\ -i\left ( f\left ( x\right ) g\left ( y\right ) \right ) ^{\frac {m}{4}}\\ i\left ( f\left ( x\right ) g\left ( y\right ) \right ) ^{\frac {m}{4}}\\ -\left ( f\left ( x\right ) g\left ( y\right ) \right ) ^{\frac {m}{4}}\end {array} \right . \]

And so on. For works for positive or negative \(n,m\) integers. Now the ode are solved each as as separable. Examples given below.