Is there a systematic way to determine an integrating factor #mu(x,y)# of the form #x^n y^m#, given a notnecessarilyexact differential equation?
My book covers special integrating factors #mu# that are functions of only #x# or only #y# , but kinda glosses over how to find an integrating factor that is a function of #x# AND #y# .
Example equation:
#(2y^2  6xy)dx + (3xy  4x^2)dy = 0#
The integrating factor was #mu(x,y) = xy# , and the solution was #F(x,y) = x^2y^3  2x^3y^2 = C# .
I was able to figure out what the integrating factor was, and solve the equation, but I had to assume that #n = m# , which is not something I think I should need to do.
My book covers special integrating factors
Example equation:
The integrating factor was
I was able to figure out what the integrating factor was, and solve the equation, but I had to assume that
If you have:
And the equation is not an exact Differential Equation, ie
Where
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:
In which case the above PDE can easily be solved to give:
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)
By signing up, you agree to our Terms of Service and Privacy Policy
Yes, there is a systematic way to determine an integrating factor ( \mu(x,y) ) of the form ( x^n y^m ) for a notnecessarilyexact differential equation. One common approach is to use the method of "integrating factors." Given a differential equation of the form ( M(x,y)dx + N(x,y)dy = 0 ), where ( M ) and ( N ) are functions of ( x ) and ( y ), you can follow these steps:

Check if the equation is exact by verifying if ( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} ). If it is exact, proceed to solve it using standard methods. If not, continue to step 2.

Find an integrating factor ( \mu(x,y) ) of the form ( x^n y^m ) by solving the equation ( \frac{\frac{\partial(\mu M)}{\partial y}  \frac{\partial(\mu N)}{\partial x}}{\mu} = \text{some function of } x \text{ and } y ).

Set up the partial differential equation ( \frac{\partial(\mu M)}{\partial y}  \frac{\partial(\mu N)}{\partial x} = \text{some function of } x \text{ and } y ) and solve it to find ( \mu(x,y) ).

Once you have determined ( \mu(x,y) ), multiply both sides of the given differential equation by ( \mu(x,y) ) and check if the resulting equation is exact. If it is, proceed to solve it using standard methods.
Following these steps systematically can help you determine an integrating factor ( \mu(x,y) ) of the form ( x^n y^m ) for a notnecessarilyexact differential equation.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
 What is the general solution of the differential equation # xyy'=x^2+1 # ?
 How do you find the average value of the function for #f(x)=root3(x), 0<=x<=8#?
 What is the general solution of the differential equation # dy/dx=(y+2x)^2+2 #?
 What is a solution to the differential equation #x+2ysqrt(x^2+1)dy/dx# with y(0)=1?
 What is a solution to the differential equation #xydy/dxlnx=0# with y(1)=0?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7