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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  1. karush

    MHB B.2.1.4 trig w/ integrating factor

    $\begin{array}{rl} \textit{Find } \mu(x): &\mu(x) =\exp\left(\int \dfrac{1}{x}\,dx\right)=e^{\ln{x}}=x\\ \textit{multiply thru by x} &xy^\prime+y=3x\cos 2x\\ \textit{rewrite as } &(xy)'=3x\cos 2x \\ \textit{}integrate &xy=\int 3x\cos 2x \...
  2. B

    Integrating Factor: Need Help Solving Excersice?

    I'm not sure if that is the right way to solve this excersice. Can someone maybe help and tell me if this calculation proofs the statement ?
  3. echomochi

    Finding an implicit solution to this differential equation

    Homework Statement Find an equation that defines IMPLICITLY the parameterized family of solutions y(x) of the differential equation: 5xy dy/dx = x2 + y2 Homework Equations y=ux dy/dx = u+xdu/dx C as a constant of integration The Attempt at a Solution I saw a similar D.E. solved using the y=ux...
  4. E

    I How do I find the integrating factor for a differential equation?

    So I've been Stuck On this Equation Trying to find the integrating factor (im not sure if it has one) appreciate the help
  5. Drakkith

    Can you find the integrating factor for this non-exact equation?

    Homework Statement Find an integrating factor of the form ##x^Ay^B## and solve the equation. ##(2y^2-6xy)dx+(3xy-4x^2)dy=0## Homework Equations ##M=2y^2-6xy## ##N=3xy-4x^2## ##IF = exp(\int \frac{M_y-N_x}{N}\,dx)## or ##IF = exp(\int \frac{N_x-M_y}{M}\,dy)## The Attempt at a Solution [/B]...
  6. binbagsss

    Is There an Integration Factor for This Differential Equation?

    Homework Statement ##cos t \frac{dv}{dt} + (sin t) t = \frac{GM}{b^2 }\sin^3 t ## Homework Equations above The Attempt at a Solution im pretty stuck to be honest. It almost looks like a product rule on the LHS but it has the wrong sign, RHS I've tried writing ##sin^3 t## as ##(1-cos^2t)\sin...
  7. H

    I How to find the integrating factor? (1st order ODE)

    x2 + y + y2dx - x dy = 0 Integrating factor, I(x,y) = -1 / (x2 + y2) How to find the integrating factor ?Why I cannot use below method to solve the ode ? (1/N)(My - Nx) = g(x) , I(x,y)=exp( ∫ g(x) dx) OR (1/M)(My - Nx) = h(y) , I(x,y)=exp( -∫ h(y) dy)
  8. karush

    MHB -206.q3.2 method of integrating factor

    2000 $\tiny{206.q3.2}\\$ $\textsf{3. use the method of integrating factor}\\$ $\textsf{to find the general solution to the first order linear differential equation}\\$ \begin{align} \displaystyle \frac{dy}{dx}+5y=10x \end{align} $\textit{clueless !}$
  9. nysnacc

    Rate of change and integrating factor

    Homework Statement Initially 5 grams of salt are dissolved in 20 liters of water. Brine with concentration of salt 2 grams of salt per lter is added at a rate of 3 liters per minute. The tank is mixed well and is drained at 3 liters a minute. How long does the process have to continue until...
  10. E

    I Integrating factor of (y+1)dx+(4x-y)dy=0

    I tried to put it in standard form as (dx/dy)+4x(1/(1+y))=y/(y+1). I get that the integrating factor is (y+1) but i am not sure if i am doing it right or what am I suppose to do next? I get (y+1) because the integrating of 1/(y+1) is ln(y+1) and since it has e, then ln cancels and i am left...
  11. chwala

    Solving a differential equation using integrating factor

    Homework Statement Solve ##{dy/dx}-2xy=2x##Homework EquationsThe Attempt at a Solution Let ##P= -2x ## and Q= 2x, Integrating factor =## e^{-x^2} ## ##y.e^{-x^2} = ∫ 2x.e^{-x^2} dx## ##y.e^{-x^2}={x^2} e^{-x^2}+∫ 2{x^3} e^{-x^2}dx## since ##y.e^{-x^2} = ∫ 2x.e^{-x^2} dx## then...
  12. EastWindBreaks

    Integrating Factor: Homework Help

    Homework Statement hello, I was reading through the textbook and I have a hard time to understand this part: Homework EquationsThe Attempt at a Solution haven't been dealing with derivatives for a while, i don't understand how it got ln |u(t)| from the first equation. Am I treating the...
  13. G

    Integrating Factor Homework: Correctness & Complexity

    Homework Statement is my integral R(x) and R(y) correct ? why it look so complicated ? Homework EquationsThe Attempt at a Solution
  14. T

    MHB Integrating Factor: Solve x ln(x) dy/dx = xe^x

    How do you solve x ln(x) dy/dx = xe^x using the integrating factor? So far, I have put it into standard form. dy/dx + y/(xln(x))=xe^x/(x(ln(x))
  15. vktsn0303

    What is an integrating factor exactly?

    While solving non-homogenous linear ODEs we make use of the integrating factor to allow us to arrive at a solution of the unknown function. Same applies to non linear ODEs where the ODEs are converted to exact differentials. But what I don't understand is how and why would someone have come up...
  16. MexChemE

    Integrating factor vs. Laplace. Engineering problems

    Hello PF! We were doing mass balances on mixing tanks in one of my ChemE courses, and in one of the problems we arrived at the following DE: \frac{dC_B}{d \theta} + 0.025C_B=0.0125 e^{-0.025 \theta} Where CB is the concetration of salt in the tank and θ is time. The professor made us solve the...
  17. Marco Lugo

    D.E. Linear equation with integrating factor

    Homework Statement https://webwork.utpa.edu/webwork2_files/tmp/equations/2d/02a7e6a06f5b2424758fa01cc965f71.png with https://webwork.utpa.edu/webwork2_files/tmp/equations/80/81c176aa8964438a63eb096513245f1.png Homework Equations [/B] Standard form: y' + p(x)y = f(x)...
  18. gfd43tg

    Integrating Factor Homework: Solving Diff. Eq w/ Constants

    Homework Statement ##C_{B}## is a function of ##\tau'##, and ##k_{1}##,##k_{2}##, and ##C_{A0}## are constants. I want to solve this differential equation \frac {dC_{B}}{d \tau'} + k_{2}C_{B} = k_{1}C_{A0}e^{-k_{1} \tau'} Homework EquationsThe Attempt at a Solution Using the integrating...
  19. _N3WTON_

    Integrating Factor Method: Finding the Solution to a Cosine Equation

    Homework Statement Find the general solution to the indicated equation: cos(x)y' + ysin(x) = 1 Homework Equations e^\int p(x)\,dx * y(x) = int\ f(x)\, dx e^\intp(x)\,dx + C The Attempt at a Solution Ok, I am having trouble getting started with this problem because I am not...
  20. C

    Use an integrating factor to solve

    Homework Statement Use an integrating factor to determine the general solutions of the following differential equation: dx/dt - 2/t = 2t3 + (4t2)(e4t) Homework Equations R(x) = e∫P(x).dx The Attempt at a Solution Usually the equation is in the form dx/dt + P(x)t = Q(x) but...
  21. J

    MHB Integrating factor, initial value problem

    $ kxy \frac{dy}{dx} = y^2 - x^2 \quad , \quad y(1) = 0 $ My professor suggests substituting P in for y^2, such that: $ P = y^2 dP = 2y dy $ I am proceeding with an integrating factor method, but unable to use it to separate the variables, may be coming up with the wrong integrating factor ( x )
  22. alane1994

    MHB Integrating factor in Exact Equations

    Ok, so I have this differential equation. \[(3x^2y+2xy+y^3)+(x^2+y^2)y\prime=0\] First I needed to check to see if it is exact. \(M=3x^2y+2xy+y^3\) \(N=x^2+y^2\) \(\dfrac{\partial M}{dy}(3x^2y+2xy+y^3)=3x^2+2x+3y^2\) \(\dfrac{\partial N}{dx}(x^2+y^2)=2x+0\) For the integrating factor, I...
  23. I

    Integrating Factor for Solving ODE with Linear Coefficients

    Problem: xy'+2y=3x Attempt: Divide by x... y'+\frac{2y}{x}=3 I think I find the integrating factor by doing: e^{\int \frac{2}{x}dx} Not sure if that's right but if it is then the solution to the integral is just 2x. Any help is appreciated
  24. 1

    Solve Linear ODE Using Integrating Factor

    Homework Statement Solve the initial value problem: $$sin(x)y' + ycos(x) = xsin(x), y(2)= \pi/2$$ Homework Equations The Attempt at a Solution Recognizing it as a Linear First-Order Equation:$$\frac{dy}{dx}+y\frac{cosx}{sinx}=x$$ $$P(x)=\frac{cosx}{sinx}$$ Integrating...
  25. T

    Solving a First ODE Using an Integrating Factor

    $$w'+2w=0\\ \frac { dw }{ dx } =-2w\\ I(x)={ e }^{ 2x }\\ \frac { dw }{ dx } { e }^{ 2x }=-2w{ e }^{ 2x }\\ \int { \frac { dw }{ dx } { e }^{ 2x } } dx=\int { -2w{ e }^{ 2x } } dx$$ Not sure what to do next.
  26. B

    Integrating Factor Homework: y'+ (2/t)y = (cost)/(t^2), y(pi)=0

    Homework Statement y' + (2/t)y = (cost)/(t^2), and the following condition is given: y(pi) = 0Homework Equations The Attempt at a Solution After employing the integrating factor, I find the solution to be: y=e^{-2t} \int e^{2t} \frac{\cos(t)}{t^2} dt. Evidently, this simplifies all the way to...
  27. Fernando Revilla

    MHB Integrating factor (Madmen's question at Yahoo Answers)

    Here is the question: Here is a link to the question: Differential equations math question? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  28. D

    Integrating factor for a 2nd order homogeneous linear ODE

    Homework Statement Consider the general linear homogeneous second order equation: P(x)y'' + Q(x)y' + R(x)y = 0 (1) We seek an integrating factor μ(x) such that, upon multiplying Eq. (1) by μ(x), we can write the resulting equation in the form [μ(x)P(x)y']' + μ(x)R(x)y = 0...
  29. P

    Differential equation parametrisation with integrating factor

    Homework Statement Use parametrisation first, derive the equation including y and p = \frac{dy}{dx} and use the integrating factor method to reduce it to an exact equation. Leave the solution in implicit parametric form. (y')^{3} + y^{2} = xyy'The Attempt at a Solution I'm really lost at...
  30. N

    Linear equations - use of integrating factor

    Hi, I am doing self study and have hit a snag in the road. Can someone please clarfiy this for me. I am reading K.A. Stroud's Engineering Mathematics which so far has been great.Consider the equation \frac{dy}{dx} + 5y = e2x In this case, we begin multiplying both sides by e5x. This gives...
  31. N

    Linear equations - use of integrating factor

    Hi, I am doing self study and have hit a snag in the road. Can someone please clarfiy this for me. I am reading K.A. Stroud's Engineering Mathematics which so far has been great. Consider the equation \frac{dy}{dx} + 5y = e2x In this case, we begin multiplying both sides by e5x. This...
  32. K

    How can I solve this without using integrating factor?

    Let the Wronskian between the functions f and g to be 3e^{4t}, if f(t) = e^{2t}, then what is g(t)? So the Wronskian setup is pretty easy W(t) = fg' - f'g = 3e^{4t} f = e^{2t} f' = 2e^{2t} So plugging it in I would get: e^{2t}g' - 2e^{2t}g = 3e^{4t} Which results in g' - 2g...
  33. W

    Integrating Factor for First Order Linear Differential Equation

    Homework Statement Find an integrating factor for the first order linear differential equation \frac{dy}{dx} - \frac{y}{x} = xe^{2x} and hence find its general solution Homework Equations The Attempt at a Solution I found the integrating factor which is e^{-lnx} = x^{-1} and...
  34. fluidistic

    First order DE with integrating factor

    Homework Statement I'm having a hard time to solve the following DE using an integrating factor, I'm asked to find one and solve the DE. (x+y^2)-2xyy'=0.Homework Equations The Attempt at a Solution If I call u=u(x) (I assume the integrating factor depends only on x, not on y), the following...
  35. B

    Solving (x^2+y)dx + (x+e^x)dy = 0 with Integrating Factor

    Homework Statement From Elementary Differential Equation by Boyce and Diprima Chapter 2 Miscellaneous Problems #11 (x^2+y)dx + (x+e^x)dy = 0 ANS:(x^3/3)+xy+e^x=c Homework Equations multiplying an integrating factor to make the DE exact: 1. du/dx = u(My - Nx)/ N 2. du/dx = u(Nx-My)/ M The...
  36. G

    Integrating factor for exact differential equation

    xdy-ydx=(x2+y2)2(xdx+ydy) (Hint: consider d(x2+y2)2)Homework Equations d(x2+y2)2 d(arctan(y/x))=xy|-y/(x2+y2)The Attempt at a Solution The answer is arctan(y/x)-(.25(x2+y2)2)=C I correctly solved other problems of this type, but this is the hardest one in the problem set and I have no idea how...
  37. L

    Derive the Integrating Factor for Homogeneous DE

    Homework Statement I have this statement: If M(x,y)dx+N(x,y)dy=0 is a homogeneous DE, then μ(x,y)=\frac{1}{xM+yN} is its integrating factor. The problem is, how do we derive this integrating factor? Homework Equations For homogeneous DE, we have f(kx,ky)=k^n*f(x,y) We also have...
  38. T

    Nonexact Differential Equation (Possible to solve by integrating factor?)

    Homework Statement Solve the differential equation: t^2 y' + y^2 = 0 The Attempt at a Solution Now, it's definitely possible to solve this via separable of variables. But I am curious to know if I can solve it with an integrating factor. Having done some reading, I noticed that this...
  39. T

    MHB Solving for P02 using Integrating Factor?

    Hello, I am solving an equation using integrating factor. I have come up to a specific point which is $$\dfrac{d}{dt} P_{02}(t) \cdot e^{(\lambda_3+\mu_3)t}=\lambda_2 \cdot P_{01}(t) \cdot e^{(\lambda_3+\mu_3)t}$$ from the previous equation, I have found $$P_{01}(t)=\lambda_1 \int_0^t...
  40. L

    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...
  41. P

    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...
  42. D

    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...
  43. H

    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...
  44. W

    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...
  45. H

    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...
  46. B

    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...
  47. C

    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...
  48. C

    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?
  49. A

    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...