\[ y^{\prime }=\frac {f\left ( x,y\right ) }{g\left ( x,y\right ) }\] ode internal name "homogeneousTypeMapleC"
This is different than the above homogeneous type C. This has the form \(y^{\prime }=\frac {f\left ( x,y\right ) }{g\left ( x,y\right ) }\) solved by transformation \(x=X+x_{0},y=Y+y_{0}\). If able to solve for \(y_{0},x_{0}\) then the ode becomes Homogeneous type A.
So what is homogeneous ode of class C ? It is an ode \(y^{\prime }=F\left ( x,y\right ) \) which is not homogeneous ode of class A but using the transformation \(x=X+x_{0},y=Y+y_{0}\) it can become one.
This means if given an ode and it is not homogeneous ode of class A then if such transformation can be found to convert it to one, it is called homogeneous ode of class C. The transformed ode is then solved in \(Y(X)\) as homogeneous ode and the solution is transformed back to \(y\left ( x\right ) \) using \(x=X+x_{0},y=Y+y_{0}\). This however required finding (if possible) the \(x_{0},y_{0}\). This section illustrates this method with an example.