# Integrating Factors

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

• Section 2.5 – Special Integrating Factors

## Expected Educational Results

• Objective 6–1: I can identify an ODE written in exact differential form.

• Objective 6–2: I can determine if an ODE is an exact equation.

• Objective 6–3: I can find the appropriate integrating factor to convert a DE in exact differential form into an exact equation.

• Objection 6–4: I can solve ODEs in exact form by applying the appropriate integrating factor.

## Integrating Factors

### Definition: Integrating Factor

If the equation $M\left(x,y\right)dx+N\left(x,y\right)dy=0$$M(x,y)dx+N(x,y)dy=0$ is not exact, but the equation $\mu \left(x,y\right)M\left(x,y\right)dx+\mu \left(x,y\right)N\left(x,y\right)dy=0$$\mu(x,y)M(x,y)dx+\mu(x,y)N(x,y)dy=0$ for $\mu \left(x,y\right)\ne 0$$\mu(x,y)\ne 0$ is exact, then the function $\mu \left(x,y\right)$$\mu(x,y)$ is an integrating factor of the equation.

#### Investigation 01

Determine if the following $\mu$$\mu$ equations are integrating factors of the DEs. Explain.

1. Is $\mu \left(x\right)=x$$\displaystyle \mu (x)=x$ an integrating factor of $\left(3{x}^{2}+y\right)dx+\left(2{x}^{2}y-x\right)dy=0$$\displaystyle (3x^2+y)dx+(2x^2y-x)dy=0$

2. Is $\mu \left(x\right)=\frac{1}{{x}^{2}}$$\displaystyle \mu (x)=\dfrac{1}{x^2}$ an integrating factor of $\left(3xy+{y}^{2}\right)dx+\left({x}^{2}+xy\right)dy=0$$\displaystyle (3xy+y^2)dx+(x^2+xy)dy=0$

3. Is $\mu \left(y\right)=\frac{1}{{y}^{4}}$$\displaystyle \mu (y)=\dfrac{1}{y^4}$ an integrating factor of $\left(2{y}^{7}+{y}^{4}\right)dx+\left(6x{y}^{6}-3\right)dy=0$$\displaystyle (2y^7+y^4)dx+(6xy^6-3)dy=0$

4. Is $\mu \left(x,y\right)=\frac{1}{{x}^{5}{y}^{2}}$$\displaystyle \mu (x,y)=\dfrac{1}{x^5y^2}$ an integrating factor of $\left({x}^{3}y-2{y}^{2}\right)dx+{x}^{4}dy=0$$\displaystyle (x^3y-2y^2)dx+x^4dy=0$