Explain why it is always possible to express any homogeneous D.E. \f$M(x, y)dx+N(x, y)dy=0\f$  in the form \f$\frac{dy}{dx}=F(\frac{y}{x})\f$  you might start by proving that: \f$M(x,y)=xM(\frac{y}{x}) \texttt{  and  }N(x,y)=x 

Any homogeneous O.D.E can be written as  \f$\frac{dy}{dx}=F(\frac{y}{x})\f$  because when identifying our functions of F(x,y) such that:  \f$\frac{\partial F}{\partial y}=M\texttt{   and   }\frac{\partial F}{\partial x}=N\f$ because our M corresponds to y and N corresponds to x. We add those two functions together and set the equal to zero and we would then get our “Total Differential”: dF=0. Then our solution would be  \f$F(x,y)=c\f$

When trying to find out if an equation is homogeneous, we must understand what an equation must look like to be homogeneous. A homogeneous differential equation is also an “exact solution,” because it has our last variable,  \f$g(x)=0\f$ .  Also worthy of note, a homogeneous differential equation will be any solution where  \f$M\neq 0\f$ .

If a function was to not be equal to 0 then the differential equation would neither be homogeneous or exact.

*Note, in most physics equations, we would change out the y variable for ‘t’ (time).