# Exact Equations

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

• Section 2.4 – Exact Equations

## Expected Educational Results

• Objective 6–1: I can identify the exact differential form.

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

• Objective 6–3: I can solve exact equations.

### Review of Partial Derivatives

You may need to review:

#### Investigation 01

Let $F\left(x,y\right)=x\mathrm{sin}\left(y\right)+{e}^{xy}-7{y}^{2}+{x}^{5}+4$$\displaystyle F(x,y)=x\sin{(y)} + e^{xy} - 7y^2 + x^5 +4$

Find the following first- and second-order partial derivatives:

• $\frac{\partial F}{\partial y}$$\displaystyle \dfrac{\partial F}{\partial y}$

• $\frac{{\partial }^{2}F}{\partial x\partial y}$$\displaystyle \dfrac{\partial^2 F}{\partial x \partial y}$

• $\frac{\partial F}{\partial x}$$\displaystyle \dfrac{\partial F}{\partial x}$

• $\frac{{\partial }^{2}F}{\partial y\partial x}$$\displaystyle \dfrac{\partial^2 F}{\partial y \partial x}$

• What do you notice about the results of the mixed second-order partial derivatives? Explain.

#### Investigation 02

1. Evaluate $\int \left(\mathrm{sin}\left(y\right)+y{e}^{xy}+5{x}^{4}\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\left(\sin{(y)}+ye^{xy}+5x^4\right)\,dx}$

2. Evaluate $\int \left(x\mathrm{cos}\left(y\right)+x{e}^{xy}-14y\right)\phantom{\rule{0.167em}{0ex}}dy$$\displaystyle \int{\left(x\cos{(y)}+xe^{xy}-14y\right)\,dy}$

3. What is the constant of integration for each of the above integrals? Explain.

4. What do you notice about the above results? Explain.

### Total Differential

#### Definition: Total Differential

Recall from Calculus III:

Suppose some function $F$$F$ is dependent on two independent variables, $x$$x$ and $y$$y$, more specifically, $F\left(x,y\right)=C$$F(x,y)=C$, where $C$$C$ is a constant. If $F$$F$ is differentiable, then the total differential is: $dF=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy$$dF=\dfrac{\partial F}{\partial x}dx+\dfrac{\partial F}{\partial y}dy$

#### Investigation 03

1. Find the total differential of ${x}^{2}y+x\mathrm{sin}\left(y\right)-x+{e}^{y}=7$$x^2y+x\sin{(y)}-x+e^y=7$.

2. Let $\frac{\partial F}{\partial x}=M\left(x,y\right)$$\displaystyle \dfrac{\partial F}{\partial x}=M(x,y)$ and $\frac{\partial F}{\partial y}=N\left(x,y\right)$$\displaystyle \dfrac{\partial F}{\partial y}=N(x,y)$.

3. Find $\frac{\partial M}{\partial y}$$\displaystyle \dfrac{\partial M}{\partial y}$

4. Find $\frac{\partial N}{\partial x}$$\displaystyle \dfrac{\partial N}{\partial x}$

5. What do you notice about the above results? Explain.