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:

• Differentiation Rules

• Integration Methods: http://www.jjw3.com/r/integrationStrategy.pdf.

Investigation 01

Let ${\displaystyle F(x,y)=x\sin (y)+e^{xy}-7y^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:

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

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

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

• ${\displaystyle {\displaystyle \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 ${\displaystyle \int (\sin (y)+y{e}^{xy}+5x^4)dx}$$\displaystyle \int{\left(\sin{(y)}+ye^{xy}+5x^4\right)\,dx}$

2. Evaluate ${\displaystyle \int (x\cos (y)+x{e}^{xy}-14y)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$$y$, more specifically, $F(x,y)=C$$F(x,y)=C$, where $C$$C$ is a constant. If $F$$F$ is differentiable, then the total differential is: $dF={\displaystyle \frac{\partial F}{\partial x}}dx+{\displaystyle \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\sin (y)-x+e^y=7$$x^2y+x\sin{(y)}-x+e^y=7$.

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

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

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

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

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Last Modified: Wednesday, 2 September 2020 07:42 EDT