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.

Method

1. Identify if the DE is exact:

   • Identify $M(x,y)$$M(x,y)$ and $N(x,y)$$N(x,y)$

   • Find ${\displaystyle \frac{\partial M}{\partial y}(x,y)}$$\displaystyle\frac{\partial M}{\partial y}(x,y)$ and ${\displaystyle \frac{\partial N}{\partial x}(x,y)}$$\displaystyle\frac{\partial N}{\partial x}(x,y)$

   • Test for exactness by verifying: ${\displaystyle \frac{\partial M}{\partial y}(x,y)=\frac{\partial N}{\partial x}(x,y)}$$\displaystyle\frac{\partial M}{\partial y}(x,y)=\frac{\partial N}{\partial x}(x,y)$

2. If the DE is not exact, then

   • if ${\displaystyle \frac{\partial M/\partial y-\partial N/\partial x}{N}}$$\displaystyle \frac{\partial M/\partial y - \partial N/\partial x}{N}$ is a function of only $x$$x$, then the integrating factor is ${\displaystyle \mu (x)=e^{\int {\displaystyle \frac{\partial M/\partial y-\partial N/\partial x}{N}}dx}}$$\displaystyle\mu(x)=e^{\int{\dfrac{\partial M/\partial y - \partial N/\partial x}{N}\,dx}}$

   • if ${\displaystyle \frac{\partial N/\partial x-\partial M/\partial y}{M}}$$\displaystyle \frac{\partial N/\partial x - \partial M/\partial y}{M}$ is a function of only $y$$y$, then the integrating factor is ${\displaystyle \mu (y)=e^{\int {\displaystyle \frac{\partial N/\partial x-\partial M/\partial y}{M}}dy}}$$\displaystyle\mu(y)=e^{\int{\dfrac{\partial N/\partial x - \partial M/\partial y}{M}\,dy}}$

   • Multiply the DE with either $\mu (x)$$\mu(x)$ or $\mu (y)$$\mu(y)$.

3. Solve the exact equation.

4. State the solution curve as $F(x,y)=C$$F(x,y)=C$.

5. Identify missing solutions.

6. Identify solutions that are not solutions of the original DE.

Investigation 03

Solve the following DEs, if possible:

1. ${\displaystyle (3x^{2}y+2xy+y^3)dx+(x^2+y^2)dy=0}$$\displaystyle (3x^2y+2xy+y^3)dx+(x^2+y^2)dy=0$

2. ${\displaystyle \frac{dy}{dx}=-\frac{3xy+y^2}{x^2+xy}}$$\displaystyle \frac{dy}{dx}=-\frac{3xy+y^2}{x^2+xy}$

3. ${\displaystyle (3x^2+y)dx+(2x^{2}y-x)dy=0}$$\displaystyle (3x^2+y)dx+(2x^2y-x)dy=0$

4. ${\displaystyle x^3\frac{dy}{dx}+4x^{2}y=x}$$\displaystyle x^3\frac{dy}{dx}+4x^2y=x$

5. ${\displaystyle (3e^{x}y+x)dx+e^{x}dy=0}$$\displaystyle (3e^xy+x)dx + e^xdy = 0$, $y(0)=1$$y(0) = 1$

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

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