2.12.1.7 Example 7 \(y^{\prime }=\frac {ay^{n}+bx^{n\left ( m+1\right ) }}{x^{nm-m+n}}\)

Solve

\[ y^{\prime }=\frac {ay^{n}+bx^{n\left ( m+1\right ) }}{x^{n\left ( m+1\right ) -m}}\]

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}\left ( \frac {ay^{n}+bx^{n\left ( m+1\right ) }}{x^{n\left ( m+1\right ) -m}}\right ) \\ & =\frac {axy^{n}+bx^{n\left ( m+1\right ) +1}}{yx^{n\left ( m+1\right ) -m}}\\ & =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 {m+1}{y}\left ( -bx^{n\left ( m+1\right ) }+ay^{n}\left ( n-1\right ) \right ) x^{-\left ( n-1\right ) \left ( m+1\right ) }\end{align*}

And let

\begin{align*} f_{y} & =y\frac {\partial F}{\partial y}\\ & =y\left ( -\frac {x\left ( ay^{n}+bx^{n\left ( m+1\right ) }\right ) }{y^{2}x^{mn-m+n}}+\frac {nxay^{n}}{y^{2}x^{mn-m+n}}\right ) \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}}\\ & =-m-1 \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

\[ F\left ( \tau \right ) =\frac {1}{\tau }\left ( a\tau ^{n}+b\right ) \]

The solution(1) becomes

\begin{align*} \ln x-c_{1}+\int ^{yx^{-\left ( m+1\right ) }}\frac {1}{\tau \left ( -\left ( -m-1\right ) -\frac {1}{\tau }\left ( a\tau ^{n}+b\right ) \right ) }d\tau & =0\\ \ln x-c_{1}+\int ^{yx^{-\left ( m+1\right ) }}\frac {1}{\tau \left ( m+1-\frac {a\tau ^{n}+b}{\tau }\right ) }d\tau & =0 \end{align*}

Which is valid solution assuming \(n,m\) are intergers.