Explain why it is always possible to express any homogeneous D.E. in the form you might start by proving that:
Any homogeneous O.D.E can be written as because when identifying our functions of F(x,y) such that: 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
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, . Also worthy of note, a homogeneous differential equation will be any solution where .
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).