# 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**

- Take the coefficient of
*dx*as*M*and the coefficient of*dy*as*N*. - Evaluate ∂
*M*/∂*y*and ∂*N*/∂*x*. - Take the difference ∂
*M*/∂*y*- ∂*N*/∂*x*. - 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. - 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. - 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. - Solve the result of Step 6 by exact equation or by inspection.