Integrating Factors

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Integrating Factors

Definition: Integrating Factor

If the equation M(x,y)dx+N(x,y)dy=0 is not exact, but the equation μ(x,y)M(x,y)dx+μ(x,y)N(x,y)dy=0 for μ(x,y)0 is exact, then the function μ(x,y) is an integrating factor of the equation.


  1. Identify if the DE is exact:

    • Identify M(x,y) and N(x,y)

    • Find My(x,y) and Nx(x,y)

    • Test for exactness by verifying: My(x,y)=Nx(x,y)

  2. If the DE is not exact, then

    • if M/yN/xN is a function of only x, then the integrating factor is μ(x)=eM/yN/xNdx

    • if N/xM/yM is a function of only y, then the integrating factor is μ(y)=eN/xM/yMdy

    • Multiply the DE with either μ(x) or μ(y).

  3. Solve the exact equation.

  4. State the solution curve as F(x,y)=C.

  5. Identify missing solutions.

  6. Identify solutions that are not solutions of the original DE.

Investigation 03

Solve the following DEs, if possible:

  1. (3x2y+2xy+y3)dx+(x2+y2)dy=0

  2. dydx=3xy+y2x2+xy

  3. (3x2+y)dx+(2x2yx)dy=0

  4. x3dydx+4x2y=x

  5. (3exy+x)dx+exdy=0, y(0)=1


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Last Modified: Monday, 6 September 2020 13:33 EDT