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. A

    How to find an integrating factor for this problem?

    Homework Statement Find an integrating factor for: xdy - (y + x^2 + 9y^2)dx = 0 Homework Equations P(x,y)dx + Q(x,y)dy = 0 Δμ = μyP - μxQ. where μ is the integrating factor. The Attempt at a Solution well, I don't know what I should do. I can use the formula I wrote but that would...
  2. G

    Integrating factor for first order linear equations uniqueness theorem

    My book stated the following theorem: If the functions P(x) and Q(x) are continuous on the open interval I containing the point x0, then the initial value problem dy/dx + P(x)y = Q(x), y(x0)=y0 has a unique solution y(x) on I, given by the formula y=1/I(x)\intI(x)Q(x)dx where I(x) is the...
  3. N

    Finding the integrating factor

    I am having trouble with the below: [ 4* (x^3/y^2) + (3/y)] dx + [3*(x/y^2) +4y]dy=0 I found My= -8x^3y^-3 - 3y^-2 and Nx= 3y^-2 i then subtracted Nx from My and divided by [3*(x/y^2) +4y] [-8x^3y^-3 - 6y^-2] / [3*(x/y^2) +4y]. can you guys give me a hint as to where my error is?
  4. A

    Integrating Factor and absolute value

    Hi, I have a general question regarding the integrating factor of first-oder linear DEs. All textbooks that I've seen (which aren't too many) simply drop the absolute symbol when the factor has the form exp(ln(abs(x))). This would evaluate to abs(x), yet the books use simply x. Why is that...
  5. F

    Integrating Factor, how do you get this?

    Homework Statement My book shows that http://img845.imageshack.us/img845/4875/unleduot.jpg and then they arrived at the results [PLAIN][PLAIN]http://img202.imageshack.us/img202/5442/unleddq.jpg So my problem is that, how exactly did they drop the mu in the first picture?
  6. C

    Exact Differential Equations and Integrating Factor

    Homework Statement "Show that each of the given differential equations of the form M(x,y)dx + N(x,y)dy = 0 are exact, and then find their general solution using integrating factors μ(x) = e∫h(x)dx and μ(x) = e∫g(y)dy Homework Equations (x2 + y2 + x)dx + (xy)dy = 0The Attempt at a Solution...
  7. P

    ODE with integrating factor NEED HELP

    Homework Statement (2x+y^2) dx +4xy dy=0,y(1)=1 Homework Equations The Attempt at a Solution I'm having trouble finding the correct integrating factor, been playing with it for an hour and have made NO progress so need help. \delta P/\delta y=2y \delta Q/\delta x=4y...
  8. J

    Deriving an Integrating Factor for a Non-Exact DE w/ Multiple Dependencies

    I was looking at how to derive an integrating factor for a non-exact DE that has multiple variable dependency, i.e. µ is xy-dependent, and I found the explanation at the link in the middle of the page at equation (22) (link...
  9. X

    Integration using integrating factor

    Homework Statement Integrate dy/dx=2y+4x+10 The Attempt at a Solution dy/dx-2y=4x+10 Integrating factor = e^(-2)dx=e^-2x multiply both sides by IF. (e^-2x)dy/dx-2y(e^-2x)=(e^-2x)(4x+10) dy/dx(e^-2x y)=(e^-2x)(4x+10) i don't know what to do next.
  10. T

    Differential equation integrating factor conceptual question.

    Homework Statement Sorry asking similar quesion again about absolute value. You can read the attachment. u(x) is the integrating factor. Why absolute value is omitted in the integration? and why the integrating factor is not "1/|x|", with the absolute sign Homework Equations The...
  11. B

    Exact Equation with Integrating Factor Help

    Homework Statement Multiply the given equation by the given integrating factor and solve the exact equation. Homework Equations ydx+(2x-yey)dy=0, \mu(x,y)=y. The Attempt at a Solution M=y2, N=2xy-y2ey Integrating N=\Psiy WRT x I get xy2-((1/3)y3ey + y2ey)+h(x)=\Psi(x,y) Differentiating...
  12. G

    Solving Integrating Factor Homework Statement

    Homework Statement use an integrating factor to solve \frac{ \partial u}{ \partial x} = -2 + \frac{u}{2x} The Attempt at a Solution let P(x) = \frac {1}{2x} M(x) = e^(\int(\frac {1}{2x}dx)) = \sqrt{x} so u = \frac{1}{ \sqrt{x}}...
  13. S

    Integrating Factor and Potential Function for Non-Exact Multi-Variable ODEs

    Hello All, Given the equation (2/y + y/x)dx + (3y/x + 2)dy I am first asked to show the equation is not exact. To do this I showed the mixed partials were not equal i.e.: (2/y + y/x)dy != (3y/x + 2)dx I am then asked to find an integrating factor and show the potential function is given...
  14. B

    Improve Your Integration Skills with Factor Homework | Solving xy' - 4y = x4ex

    Homework Statement xy' - 4y = x4ex Homework Equations The Attempt at a Solution y' - 4x-1y = x3ex x-4y' - 4x-5y = x-1ex I'm not sure what to do next, I can't express the LS as a derivative
  15. A

    Integrating Factor Greetings: What is it & Why Use It?

    greetings what does a integrating factor tells about a differential equation? in order to find the solution for a exact equation we multiply the equation by integrating factor(I.F). as intergrating factor=e^integration(p)dx i.e given by I.F=e^gx where gx is integration of p now as we have...
  16. H

    Inexact DGL, integrating factor

    Hello I'm trying to solve the following DGL with an integrating factor: x'=xg(y) y'=yh(x) which is equivalent to -yh(x)dx+xg(x)dy=0 which is an inexact dg? How to i find an integrating factor in this case? thx
  17. T

    Differential equations question - possible integrating factor?

    Homework Statement Solve the differential equation with the initial condition y(0) = 1 Homework Equations 3e5xdy/dx = -25x/y2 The Attempt at a Solution First I tried putting everything with an x or dx on one side and a y or dy on the other side, and solved for C. I got 0 as the...
  18. Saladsamurai

    Integrating Factor (Derivation)

    Hello all! :smile: I through a section in my text (by https://www.amazon.com/dp/0133214311/?tag=pfamazon01-20) on 1st Order linear ODEs. I am understanding the derivation of the integrating factor method pretty well; however, there are some aspects of the mathematics that I am getting hung up...
  19. L

    First order nonlinear ODE - Integrating factor + exact differentials, or not?

    First order nonlinear ODE -- Integrating factor + exact differentials, or not? Hello everyone, (I apologize if this did not format properly, if not I will attempt to edit it if that functionality is available upon submitting a question). I recently came across the following nonlinear ODE...
  20. M

    1st order linear differential eq. using integrating factor

    Homework Statement Solve the inital value problem for y(x); xy′ + 7y = 2x^3 with the initial condition: y(1) = 18. y(x) = ? Homework Equations dy/dx +P(x)y=Q(x), integrating factor=e^∫P(x) dx The Attempt at a Solution Multiplied all terms by 1/x to get it in correct form...
  21. D

    Solve DE by finding an integrating factor (please check my work)

    Homework Statement Solve (x + 2) sin y dx + x cos y dy = 0 by finding an integrating factor .....M......N..... M_y (x+2)siny = (x+2)cosy N_x xcosy = cos y M_y not equal to N_x, therefore equation is not exact.. so far so good? thanks Homework Equations The Attempt at...
  22. D

    Verifying an integrating factor -please check my work , thanks.

    Verifying an integrating factor --please check my work , thanks. Homework Statement Verify that 1/y^4 is an integrating factor for (3x^2-y^2)dy/dx - 2xy = 0, and then use it to solve the equation (1/y^4)* (3x^2-y^2)dy/dx = 2xy* (1/y^4) =3x^2 / y^4 dy = 2x/y^3 dx =3x^2 / y dy = 2x...
  23. D

    Can't determine the correct integrating factor (DE)

    Homework Statement Solve (x+2)sin y dx + xcos y dy = 0 by finding integrating factor mu(x) Homework Equations i put equation in standard form to get integrating factor dy/dx + (x+2)sin y / x cos y = 0 but don't i need to get the y out of the sin and cos somehow? so that...
  24. D

    Verify an integrating factor for differential equation stuck

    Homework Statement verify that mu= 1/(y^4) is an integrating factor of: (3x^2 - y^2)dy/dx - 2xy = 0 and use it to solve the equation. Homework Equations (3x^2 - y^2)dy/dx - 2xy = 0 first i want equation in standard form right? dy/dx - 2xy / (3x^2 - y^2) = 0 then mu =...
  25. D

    Integrating factor problems (property of exponents)

    Homework Statement my integrating factor for the DE ty' + (t+1)y = t is mu(x) = e^integ (1+(1/t)) = e^(t + ln|t| + c) so does this simplify to this...or not? = e^t + t + c so that DE becomes: ((e^t) + t))y = (e^t) + t) and then after integrating... ((e^t) + t))y = e^t +...
  26. M

    Solving ODE with Integrating Factor Method: (1-x/y)dx + (2xy+x/y+x^2/y^2)dy = 0

    Homework Statement Solve: (1-\frac{x}{y})dx + (2xy + \frac{x}{y} + \frac{x^2}{y^2})dy = 0 The Attempt at a Solution No idea what strategy to use here. Tried using an integrating factor, but no success. A lot of x/y in here makes me think I need to use a substitution, but there's also...
  27. C

    Ordinary DE using integrating factor?

    Homework Statement solve the following initial condition problem x (d/dx) y(x) + 4xy(x) = -8-y(x) y(4)=-6 Homework Equations The Attempt at a Solution first i rearranged xy'+y+4xy=-8 xy'+y(1+4x)=-8 y'+y(1/x+4)=-8/x integrating factor:e^\int(1/x+4) e^(lnx+4x) x+e^4x multiplying...
  28. N

    Find integrating factor and solve the equation 3

    y' + y = e^x ; y(0) = 1 1st, i calculate the integrating factor... u(x) = e^x times the integrating factor with DE... y'e^x + ye^x = e^2x dy/dx e^x + ye^x = e^2x d/dx ye^x = e^2x ye^x = ∫ e^2x dx ...= 1/2 e^2x + C y = 1/2 e^x + C the problem here, i didn't get the...
  29. N

    Find integrating factor and solve the equation 2

    Question: (1 - x - z)dx + dz = 0 let M = (1 - x - z) ; N = 1 dM/dz = -x ; dN/dx = 0 hence, not exact. integrating factor... 1/N[ dM/dz - dN/dx ] = 1[-x] .......= -x e^∫-x dx = e ^ [(-x^2)/2] times integrating factor with DE.. {e ^ [(-x^2)/2] -x e ^ [(-x^2)/2] - z e ^...
  30. N

    Find integrating factor and solve the equation

    y dx + x ln x dy = 0 ; x > 0 my integrating factor is x.. so.. multiply with DE, xy dx + x^2 ln x dy = 0 let M = xy ; N = x^2 ln x dM/dy = x ; dN/dx = x + 2x ln x the problem is.. i didn't get the exact equation after multiply the integrating factor.. I've double...
  31. N

    Differential equation given integrating factor

    Homework Statement Show that given function μ is an integrating factor and solve the differential equation.. y^2 dx + (1 + xy) dy = 0 ; μ(x) = e^xy The Attempt at a Solution let M = y^2 N = (1 + xy) dM/dy = 2y dN/dx = y hence, not exact equation. times μ(x) = e^xy to the...
  32. P

    First-Order integrating factor of the form f(xy)

    (Moderator's note: thread moved from "Differential Equations") M(x,y) + N(x,y)(dy/dx) = 0 f'(xy) = G(xy)f(xy) where G(xy) = (Nx - My)/(xM - yN) Replace xy with a single variable to obtain a simple 1st order differential equation and find f(xy). I got to: ln|f| = Integral(G(xy)) by...
  33. J

    Integrate Factor Diff. Equation | Step-by-Step Solution

    Integrating factor! As promised I'm back with integrating factor differential equation.(x^2 + 1)dy/dx -2xy = 2x(x^2+1) y(0)=1 First put into standard from by dividing thru by (x^2 +1 )dy/dx -2xy/(x^2 + 1) = 2x Integrating factor is given by exp( integral of -2x(x^2 + 1))...
  34. S

    Separation of variables / integrating factor problem

    Homework Statement Find the general solution for this differential equation. dy −2x^2 + y^2 + x dx = x yHomework Equations The Attempt at a Solution dy/dx = (−2x^2 + y^2 + x) / (x y) let y^2 = v dy/dx = v + x dv/dx v + x (dv/dx) = (-2x^2 + v^2 x^2 + x ) / v x^2 => x (dv/dx)...
  35. P

    ODE question - Integrating factor

    Homework Statement Find all solutions of the equation: y' = (2y)/(t.logt) = 1/t, t > 0 Homework Equations Integrating factor I = exp(\intp(x)dx) where y' + p(x)y = q(x) The Attempt at a Solution Hi everyone, here's what I've done so far: Let p(t) = -2/(t.logt) I =...
  36. B

    Why e-Integral Factor Omits Constant Integral?

    whenever i see that integrating factor for solving a linear differential equation with eint. p(x) dx and then multiplied out in the equation, there seems to be no constant. i tried solving an equation with it the other day, and got an incorrect solution because of it (i think. at least i got a...
  37. James889

    First order DE, using integrating factor

    Hi, I tried to solve this by using the integrating factor technique \begin{cases} dy/dt +10y = 1 \\ y(1/10) = 2/10 \end{cases} So p(x) = 10t \rightarrow e^{10t} e^{10t} \cdot \frac{dy}{dt} + e^{10t} \cdot 10y = e^{10t} This part is confusing to me, i have two different variables...
  38. D

    Integrating factor problems (differential equation with initial conditions)

    Homework Statement Find general solution of equation (t^3)y' + (4t^2)y = e^-t with initial conditions: y(-1) = 0 and t<0 book answer gives y = -(1+t)(e^-t)/t^4 t not = 0 Homework Equations The Attempt at a Solution (t^3)y' + (4t^2)y = e^-t get integrating...
  39. M

    Integrating Factor Method and Absolute Value Bars

    So there's this equation: x^2 y^2 dx + (x^3y-1)dy It has to be solved with the integrating factor method, so I get this: \mu(y) = e^{\int \frac{dy}{y}} = e^{\ln{|y|}} = |y| My question is, what do I do with the absolute value bars? If I just drop them and multiply the entire equation with...
  40. N

    Finding the integrating factor (ODEs)

    Finding the integrating factor (ODEs) [Solved] Working on this problem, I can't figure out why we take the derivative of \mu with respect to y, and what to do when our integrating factor is a function of both x and y. In the case below, it ended up being separable, but what can you do if it's...
  41. E

    Holonomic constraints integrating factor question

    Ok, here's the question, have patience with my terrible latex skills... Homework Statement The equations of constraint of the rolling disk: dx - asin(theta)d(phi) = 0 -> 1. dy + acos(theta)d(phi) = 0 -> 2. are special cases of general linear diff-eqs of constraint of...
  42. I

    What is the Integrating Factor for Solving this Differential Equation?

    Homework Statement Solving this differential equation ty' + 2y = t^2 - t + 1 Homework Equations Its linear so i set it up in linear form y' + y(2/t) = t - 1 +1/t The Attempt at a Solution the integrating factor (u) = e ^ integral (ydt) = e^ integral (2dt/t)... Here's the...
  43. CFDFEAGURU

    Differential Equation using Integrating Factor

    Homework Statement The ODE is (1+x)*dy/dx - xy = x+x^2 Homework Equations The method of solution is to be through the use of the integration factor. The Attempt at a Solution First, I divided each side by (1+x) to produce dy/dx - xy/(1+x) = x then factor out the x on...
  44. K

    Existence of integrating factor

    During one lecture it was mentioned that equations of the form P(x,y)dx+Q(x,y)dy=0 always have at least one integrating factor. But the lecturer didn't know the proof, I've tried using Google but no luck. Anybody can show me the proof? Thanks a lot.
  45. S

    Integrating factor in Entropy (1/T)

    Homework Statement Show that the statement for Entropy dS = \int\frac{\delta Q}{T} is path independent Homework Equations The Attempt at a Solution I am trying to show this by stating that dS is an exact differential by stating how \delta Q is an inexact differential and by multiplying by...
  46. S

    Problem with integrating factor

    what next ? TNX !
  47. Wellesley

    Integrating Factor for Exact Differential Equation

    I can see why y2 is an integrating factor, but I don't know how this answer can be derived. Do you use: \frac{N_{x}(x,y) - M_{y}(x,y)}{M(x,y)} Where M (x,y) = 4(x3/y2)+(3/y)]dx, and N = 3(x/y2)+4y dy When I try and solve this, I get a very complex integrating factor...
  48. R

    Integrating Factor: -g(m/g) and its Derivative

    In the following problem how does -g end up becoming -g(m/g)? Isn't the derivative of -g just (-g^2/2)? http://users.on.net/~rohanlal/integfact.jpg
  49. tony873004

    ODE, Integrating Factor method

    I'm trying to follow this example from class notes. x^2 \frac{{dy}}{{dx}} + xy = 1,\,\,y\left( 1 \right) = 2 Divide through by x^2 to set it up in a form for using the integrating factor method. \frac{{dy}}{{dx}} + \frac{y}{x} = - \frac{1}{{x^2 }} I'm not sure where the minus...
  50. S

    Integrating Factor and Unintegratable Term

    I was given the following ODE to solve and it seemed simple enough. However, after you have used the integrating factor the integral is not integratable. y' = (1+x^2)y +x^3, y(0)=0 Find y(1) if y(x) is the solution to the above ODE. So I put it in the proper form of: y' + (-1-x^2)y...
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