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(x,y)dx+N(x,y)dy=0$$M(x,y)dx+N(x,y)dy=0$ is not exact, but the equation $\mu (x,y)M(x,y)dx+\mu (x,y)N(x,y)dy=0$$\mu(x,y)M(x,y)dx+\mu(x,y)N(x,y)dy=0$ for $\mu (x,y)\ne 0$$\mu(x,y)\ne 0$ is exact, then the function $\mu (x,y)$$\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 ${\displaystyle \mu (x)=x}$$\displaystyle \mu (x)=x$ an integrating factor of ${\displaystyle (3x^2+y)dx+(2x^{2}y-x)dy=0}$$\displaystyle (3x^2+y)dx+(2x^2y-x)dy=0$

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

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

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

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