The Determination of Integrating Factor

From the differential equation
 

$M ~ dx + N ~ dy = 0$

 

Rule 1
If   $\dfrac{1}{N}\left( \dfrac{\partial M}{\partial y} - \dfrac{\partial N}{\partial x} \right) = f(x)$,   a function of x alone, then   $u = e^{\int f(x)~dx}$   is the integrating factor.

 

Rule 2
If   $\dfrac{1}{M}\left( \dfrac{\partial M}{\partial y} - \dfrac{\partial N}{\partial x} \right) = f(y)$,   a function of y alone, then   $u = e^{-\int f(y)~dy}$   is the integrating factor.

 

Note that the above criteria is of no use if the equation does not have an integrating factor that is a function of x alone or y alone.
 

Steps

  1. Take the coefficient of dx as M and the coefficient of dy as N.
  2. Evaluate M/y and N/x.
  3. Take the difference M/y - N/x.
  4. Divide the result of Step 3 by N. If the quotient is a function of x alone, use the integrating factor defined in Rule 1 above and proceed to Step 6. If the quotient is not a function of x alone, proceed to Step 5.
  5. Divide the result of Step 3 by M. If the quotient is a function of y alone, use the integrating factor defined in Rule 2 above and proceed to Step 6. If the quotient is not a function of y alone, look for another method of solving the equation.
  6. Multiply both sides of the given equation by the integrating factor u, the new equation which is uM dx + uN dy = 0 should be exact.
  7. Solve the result of Step 6 by exact equation or by inspection.

 

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