Is there a systematic way to determine an integrating factor ##mu(x,y)## of the form ##x^n y^m##, given a not-necessarily-exact differential equation?

If you have:

##M(x,y) dx + N(x,y)dy=0##

And the equation is not an exact Differential Equation, ie

##(partial M) /(partial y) != (partial N) / (partial x) ##

Then you must convert the equation into an exact differential equation by multiplying by an integrating factor ##mu(x,y)## to get

##mu(x,y)M(x,y) dx + mu(x,y)N(x,y)dy=0##

Where

##(partial (muM)) /(partial y) = (partial (muN)) / (partial x) ##

That's all well and good but In order to find such an integrating factor ##mu(x,y)## you can do some manipulation and eventually establish the need to solve the partial differential equation:

## " " M (partial mu) /(partial y) - N(partial mu) /(partial x) + ((partial M) /(partial y) - (partial N) /(partial x) ) mu ## = 0 or, ## " " Mmu_y-Nmu_x + (M_y-N_x)mu=0##

which in general is a harder problem to solve!

If the given differential equation is "designed" to be solved (eg in an exam rather than a real life equation) then it will often be the case that:

either ## " " mu(x,y) = mu(x)##, a function of ##x## alone or ## " " mu(x,y) = mu(y)##, a function of ##y## alone

In which case the above PDE can easily be solved to give:

## " " mu(y) = exp(int ((partial M) /(partial y)-(partial N) /(partial x))/M dy) = e^(int (M_y-N_x)/M dy)## or ## " " mu(x) = exp(int ((partial M) /(partial y)-(partial N) /(partial x))/N dx) = e^(int (M_y-N_x)/N dx)##

respectively.

But, in general finding the integrating factor will not be possible and so the Differential Equation would be solved numerically rather than finding an analytical solution.

In the real world, It is always possible to find a series solution but this approach is particularly cumbersome (and is often the approach used by a computer for a numerical solution)