3.3.15 Polynomial ode \(y^{\prime }=\frac {a_{1}x+b_{1}y+c_{1}}{a_{2}x+b_{2}y+c_{2}}\)

ode internal name "polynomial"

Special form for first order ode where the lines \(a_{1}x+b_{1}y+c_{1}=0,a_{2}x+b_{2}y+c_{2}=0\) can be either parallel or not parallel. If the lines are not parallel then the transformation \(X=x-x_{0},Y=y-y_{0}\) transforms the ode to homogeneous ode. If the lines are parallel then the transformation \(U\left ( x\right ) =a_{1}x+b_{1}y\) converts the ode to separable in \(U\left ( x\right ) \). The not parallel case is when \(\frac {a_{1}}{b_{1}}\neq \frac {a_{2}}{b_{2}}\) and the second case is when \(\frac {a_{1}}{b_{1}}=\frac {a_{2}}{b_{2}}\).