Author: John J Weber III, PhDCorresponding 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 $M(x,y)dx+N(x,y)dy=0$ is not exact, but the equation $\mu (x,y)M(x,y)dx+\mu (x,y)N(x,y)dy=0$ for $\mu (x,y)\ne 0$is exact, then the function $\mu (x,y)$ is an integrating factor of the equation.

Investigation 01

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

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

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

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

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