Explain why it is always possible to express any homogeneous D.E.

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$$ . Also worthy of note, a homogeneous differential equation will be any solution where $\f$M\neq 0\f$$ .