# What is Integrating factor: Definition and 123 Discussions

In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field). This is especially useful in thermodynamics where temperature becomes the integrating factor that makes entropy an exact differential.

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41. ### How Do You Find Integrating Factors for a Given Differential Equation?

Homework Statement Hi, I attempted to solve this question, but it did not work well. I would be grateful if you could help me. Show that an equation of the form xrys(mydx + nxdy)= 0 has an infinite number of integrating factors of the form xayb, and find expressions for a and b...
42. ### Integrating Factor Proof: Combining Equations for Exact First-Order ODEs

Hey, I've just been following this proof for a integrating factor of (xy), http://mathworld.wolfram.com/ExactFirst-OrderOrdinaryDifferentialEquation.html it starts at at equation (22) I understood it all a few days ago and now I seem to have forgotten this one step. It says...
43. ### Integrating Factor Differential Equation

Homework Statement e∫^P(x) ∫\frac{x-2}{x(x-1)}dx The Attempt at a Solution so i split it into ∫\frac{x-2}{x(x-1)}dx = ∫\frac{2x-1}{x^2-x}dx - ∫\frac{x+1}{x^2-x}dx = ln(x2-x) - ∫\frac{x}{x^2-x} - ∫(x2-x)-1 = ln(x2-x) - ln(x-1) - ∫(x2-x)-1 ok. having problems working out...
44. ### What is the Correct Way to Use the Integrating Factor Method for Proving an ODE?

I am really struggling with proving a ODE by means of using the integrating factor method. My original problem was a Laplace transform q'+2q=5sin(t) where q(0)=0 I believe i have got the correct naswer for this as being:- q= e^-2t +2sint-cost I just need to confirm this i have my...
45. ### Solving a first order DE using integrating factor method

Homework Statement (3(x^2)y + 2xy + y^3)dx + (x^2 + y^2)dy = 0 The Attempt at a Solution This is from my notes, so I already have the answer. I just don't understand the very last step with the integrations. (3(x^2)y + 2xy + y^3)dx + (x^2 + y^2)dy = 0 My = 3x^2 + 2x + 3y^2 Nx = 2x...
46. ### Help with ODE using an integrating factor

Hi all, I am doing some Laplace Transforms as part of my HND, i have got an answer for this question q' +2q = 5sint q(0)=0, t(0)=0 But i need to prove it by means of using an integrating factor method. My original answer is:- e^-2t +2sint-cost does this look right? I also have...
47. ### Integrating Factor Method Problem

Homework Statement Find the general solution of the given differential equation cosxy'+(sinx)y=1 The Attempt at a Solution I divided everything by cosx and got : y'+(tanx)y=secx then after doing e to the integral of tanx i got : ∫d/dx[secx*y]=∫secx after integrating and...
48. ### Using an integrating factor properly

alright guys, I've been trying to tackle this for a couple of hours now. dy/dt-2y=4-t my integrating factor is e^(-2t) of course. dy(e^(-2t))/dt-2ye^(-2t)=4e^(-2t)-te^(-2t) then I get completely lost. how do I integrate when it's like this? My book simplifies the above equation into...
49. ### IVP Solving with Integrating Factor: Finding a Solution

I keep struggling to find a solution to this IVP. We are supposed to use integrating factors y'-(1/t)y=8t^2+te^t t>0, y(1)=6 I get an integrating factor of (1/t) and general solution of y=4t^3+te^t+c but then i get e+2 for c. This doesn't seem correct to me, any suggestions?
50. ### A challenging ODE question: find an integrating factor

This is the ODE: y' + siny + xcosy + x = 0. The problem is: Find an integrating factor for the ODE above. You can see my solution to the ODE here: https://www.physicsforums.com/showthread.php?t=543662. from my solution it seems that e^x(sec^2(y/2)) must be an integrating factor. but I fail...