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.

Method

1. Identify if the DE is exact:

• Identify $M\left(x,y\right)$$M(x,y)$ and $N\left(x,y\right)$$N(x,y)$

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

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

2. If the DE is not exact, then

• if $\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 $\mu \left(x\right)={e}^{\int \frac{\partial M/\partial y-\partial N/\partial x}{N}\phantom{\rule{0.167em}{0ex}}dx}$$\displaystyle\mu(x)=e^{\int{\dfrac{\partial M/\partial y - \partial N/\partial x}{N}\,dx}}$

• if $\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 $\mu \left(y\right)={e}^{\int \frac{\partial N/\partial x-\partial M/\partial y}{M}\phantom{\rule{0.167em}{0ex}}dy}$$\displaystyle\mu(y)=e^{\int{\dfrac{\partial N/\partial x - \partial M/\partial y}{M}\,dy}}$

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

3. Solve the exact equation.

4. State the solution curve as $F\left(x,y\right)=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. $\left(3{x}^{2}y+2xy+{y}^{3}\right)dx+\left({x}^{2}+{y}^{2}\right)dy=0$$\displaystyle (3x^2y+2xy+y^3)dx+(x^2+y^2)dy=0$

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

3. $\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$

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

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