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. Y

    Integrating factor for solving equation problem.

    Homework Statement Find the gerneral solution of the differential equation below: dy/dt=(-y/t)+2 Homework Equations none The Attempt at a Solution my solution by using integrating factor: 1.find the homogenous solution first dy/y = -1/t dt you get ln(y) = -ln(t) when...
  2. H

    Finding the Integrating Factor

    Hi ok so te question is asking for me to fin the integrating factor of (y2x+y)dy + (x2y+2x)dy = 0 the only thing i know is that the integrating factor should be a function of xy Can some one pleas explain how to do this? Thank you.
  3. J

    What is purpose of an integrating factor and how do you get one?

    Homework Statement Am I doing this correctly? Is 2y^4dx +4xy^3dy exact or inexact? Homework Equations The Attempt at a Solution So if its exact, then M(x,y)dx = N(x,y)dy right? (Euler) But how can I take partial derivative of 2y^4 with respect to x, if there is no x to...
  4. R

    Solving Differential Equations with Integrating Factor

    Hello, I was looking at the differential equation (x^2 + 1) \frac{dy}{dx} + xy = 0. This solution to this equation has to be y = \frac{c}{\sqrt{x^2 +1}} So, when do we usually need to use the method of integrating factor? Can we solve all linear equations (1st order) using this method...
  5. P

    Easier way of finding Integrating Factor for Exact Differential Equation?

    Easier way of finding Integrating Factor for Exact Differential Equation?? Homework Statement ,find an integrating factor and then solve the following: [4(x3/y2)+(3/y)]dx + [3(x/y2)+4y]dy = 0 Homework Equations u(y)=y2 is a valid integrating factor that yields a solution...
  6. J

    Is -t the Correct Integrating Factor for this Diff Eq?

    [SOLVED] Integrating Factor Diff Eqs Homework Statement g ' - g/t = t e^t I'm trying to solve this, but I seem to have run into a problem, according to my book the integrating factor is 1/t, however I believe that it is -t Homework Equations The Attempt at a Solution...
  7. E

    Understanding Integrating Factor for Linear ODEs

    Hi, I'm learning differential equations, and although I understand the methods I have learned thus far, I often have trouble seeing what is the reasoning behind them. Take for example, the use of the integrating factor when solving first order linear ODE's. I understand how to use it, but...
  8. S

    Special Integrating Factor?

    1. Solve the equation [sin(xy) + xycos(xy)]dx + [1 +x^{2}cos(xy)dy = 0 Homework Equations The Attempt at a Solution [sin(xy) + xycos(xy)]dx = M(x,y) [1 +x^{2}cos(xy)dy = N(x,y) dM/dy sin(xy) = xcos(xy) xycos(xy) = uv' + vu' = (xy)(-xsin(xy)) + (x)(cos(xy)) u = xy du = x...
  9. B

    Solve 1st Order Diff Eq Using Integrating Factor: 2y/x + dy/dx = 1-2/x

    Homework Statement Use the integrating factor to solve the first order differential equation dy/dx + (2/x)y = 1 - 2/x subject to the initial condition y(1) = 2The Attempt at a Solution I have the full answer (it's an example), but my lecturer is notorious for omitting steps, to my constant...
  10. B

    What is the issue with the two methods used to solve the ODE dy/dx=x(1-y)?

    Got the eqn dy/dx=x(1-y) and it can be solved both linear and separable methods.(Linear method being using a integrating factor) Problem I am having is that with this two methods i get two different (yet similar answers) and was wondering if you can see my problem with this two methods I am...
  11. M

    Need Help with Differential Equation (integrating factor)

    Hi there, I am having a bit of difficulty finding the integration factor for the following problem. The problem lies in taking the integral of a function of two variables. Anyways, here's what I have: y^2+y-xy'=0 I then divided by x (i prefer it this way), so \frac{y^2+y}{x} + y' = 0 then...
  12. D

    Why Do I Need an Integrating Factor for Polar Coordinates?

    I'm solving the following equation in the unit square using finite differences: epsilon(u_xx+u_yy)+u_x+u_y=0, where epsilon is a singular perturbation parameter. I need to use domain decomposition to isolate the corner singularity in the outflow corner. My subdomain in this corner is a...
  13. G

    Solving DE: Integrating Factor Method

    What approach should be used to solve the following DE: dy/dx= (-2x+5y)/(2x+y) Find an integrating factor and solve it as an excact equation? Thanks.
  14. M

    Solving Non-Exact Differential Equations with Integrating Factor

    Hello everyone I understand how to solve exact equations, but what happens when they arnt' exact? I'm confused on what I'm suppose to do! Does anyone feel like explaning hte process to me, if given an integrating factor/> or give me a website? Here is my problem: Check that the equation...
  15. B

    What is the Integrating Factor for x^n*y^m?

    Hey everyone, I need to find an integrating factor of the form x^n*y^m, to solve a differential equation i have... however i do not know the process to solve for an integration of this form.. .any help?? Thanks Steph
  16. A

    Integrating factor of (a+1)ydx + (b+1)xdy = 0

    for the equation, (a+1)ydx+(b+1)xdy=0, i am wondering how to get (x^a)(y^b) as an integrating factor~ the following is my work: (1/F)(dF/dx)=(a-b)/[(b+1)x] => F=cx^[(a-b)/(b+1)] why doesn't that method work?
  17. A

    Is e^x the Optimal Integrating Factor for Solving Differential Equations?

    for the question, siny+cosydy=0, i want to find an integrating factor. my work: (1/F)(dF/dx)=(1/cosy)(cosy+siny)=1+tany =>lny=x +xtany +c` => y =ce^(x+xtany) however, the question wants the integrating factor to be e^x... why?
  18. R

    Solving ODE: Integrating Factor for Problem 4d

    Howdy, I've read this forum for some time, however this is my first post. I am attempting to solve this ODE. I am looking to find an integrating factor, then solve. I have attached the link to the problem set if my input here is ambiguous. Number 4d. Thank you kindly for any help you might...
  19. B

    Solving "ydx - \left( {x + y^3 } \right)dy = 0" with an Integrating Factor

    Q. By finding a suitable integrating factor, solve the following equation: \left( {x + y^3 } \right)y' = y (treat y as the independent variable). Answer: Exact equation is y^{ - 1} \left( {\frac{{dx}}{{dy}}} \right) - xy^{ - 2} = y leading to x = y\left( {k + \frac{{y^2 }}{2}} \right)...
  20. himanshu121

    How do we find Integrating factor for a General Diff equation

    For eg is there a way to find IF for pydx +qxdy +x^my^n(rydx+sxdy)=0
  21. N

    Integrating factor strategy

    Q. Motivate the Integrating factor strategy for U ( "Mew" ) of y I know how to prove it for "Mew" of x but how to do for "mew" of y Maybe something like this. Mdx (x.y) + Ndy ( x, y ) = 0 Assume this is differentiable so let us multiply by "mew" of x on both sides to make it...
  22. N

    Need Help - Exact Integrating Factor?

    Here's the problem: y\left( 2x-y-1 \right) dx + x\left( 2y-x-1 \right) dy = 0 So rewrite it as: \left( 2xy-y^2-y \right) dx + \left( 2xy-x^2-x \right) dy = 0 Now it's in the form,…P(x,y) dx + Q(x,y) dy = 0 \frac{\partial P(x,y)}{\partial y} = 2x-2y-1 \frac {\partial...
  23. Math Is Hard

    Solving a Linear Equation with e^(x^2) Integrating Factor

    I am trying to solve a linear equation and getting stuck. y' + 2xy = x^2 I am using e^{x^2} as my integrating factor and multiplying that to both sides. Afterwards, I am able to wrap up the LHS as [y e^{x^2}]' and I have [y e^{x^2}]' = x^2 e^{x^2} Now all I need to do is...
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