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.

Exact Equations

Method of Solving Exact Equations

1. Identify $M(x,y)$$M(x,y)$ from the exact differential form.

2. Identify $N(x,y)$$N(x,y)$ from the exact differential form.

3. Find ${\displaystyle \frac{\partial M(x,y)}{\partial y}}$$\displaystyle \frac{\partial M(x,y)}{\partial y}$.

4. Find ${\displaystyle \frac{\partial N(x,y)}{\partial x}}$$\displaystyle \frac{\partial N(x,y)}{\partial x}$.

5. Compare ${\displaystyle \frac{\partial M(x,y)}{\partial y}}$$\displaystyle \frac{\partial M(x,y)}{\partial y}$ and ${\displaystyle \frac{\partial N(x,y)}{\partial x}}$$\displaystyle \frac{\partial N(x,y)}{\partial x}$.

   • If ${\displaystyle \frac{\partial M(x,y)}{\partial y}=\frac{\partial N(x,y)}{\partial x}}$$\displaystyle \frac{\partial M(x,y)}{\partial y}=\frac{\partial N(x,y)}{\partial x}$, then the differential equation is an exact equation, proceed to next step.

   • If ${\displaystyle \frac{\partial M(x,y)}{\partial y}\ne \frac{\partial N(x,y)}{\partial x}}$$\displaystyle \frac{\partial M(x,y)}{\partial y}\ne\frac{\partial N(x,y)}{\partial x}$, then we need another method to be discussed in a future class period.

6. Since we know partial derivatives of $F(x,y)$$F(x,y)$, we need to integrate the partial derivatives to find $F(x,y)$$F(x,y)$:

   • Since ${\displaystyle M(x,y)=\frac{\partial F(x,y)}{\partial x}}$$\displaystyle M(x,y)=\frac{\partial F(x,y)}{\partial x}$, to find $F(x,y)$$F(x,y)$ evaluate: ${\displaystyle \int M(x,y)dx}$$\displaystyle \int{M(x,y)}\,dx$, OR

   • Since ${\displaystyle N(x,y)=\frac{\partial F(x,y)}{\partial y}}$$\displaystyle N(x,y)=\frac{\partial F(x,y)}{\partial y}$, to find $F(x,y)$$F(x,y)$ evaluate: ${\displaystyle \int N(x,y)dy}$$\displaystyle \int{N(x,y)}\,dy$

7. Since we are evaluating indefinite integrals, the antiderivatives require a constant term:

   • When integrating with respect to $x$$x$, $y$$y$ is a constant; evaluate ${\displaystyle F(x,y)=\int M(x,y)dx+g(y)}$$\displaystyle F(x,y)=\int{M(x,y)}\,dx+g(y)$; OR

   • When integrating with respect to $y$$y$, $x$$x$ is a constant; evaluate ${\displaystyle F(x,y)=\int N(x,y)dy+g(x)}$$\displaystyle F(x,y)=\int{N(x,y)}\,dy+g(x)$

8. Find the constant of integration:

   1. Case 1

      • Evaluate: ${\displaystyle \frac{\partial F(x,y)}{\partial y}=\frac{\partial [\int M(x,y)dx+g(y)]}{\partial y}=M(x,y)+g^{\prime }(y)}$$\displaystyle \frac{\partial F(x,y)}{\partial y}=\frac{\partial \left[\int{M(x,y)}\,dx+g(y)\right]}{\partial y}=M(x,y)+g^{\,\prime}(y)$ [which is equivalent to $N(x,y)$$N(x,y)$]

      • Solve $M(x,y)+g^{\prime }(y)=N(x,y)$$M(x,y)+g^{\,\prime}(y)=N(x,y)$ for $g^{\prime }(y)$$g^{\,\prime}(y)$

      • ${\displaystyle g(y)=\int g^{\prime }(y)dy}$$\displaystyle g(y) = \int{g^{\,\prime}(y)\,dy}$

   2. Case 2

      • Evaluate: ${\displaystyle \frac{\partial F(x,y)}{\partial x}=\frac{\partial [\int N(x,y)dy+g(x)]}{\partial x}=N(x,y)+g^{\prime }(x)}$$\displaystyle \frac{\partial F(x,y)}{\partial x}=\frac{\partial \left[\int{N(x,y)}\,dy+g(x)\right]}{\partial x}=N(x,y)+g^{\,\prime}(x)$ [which is equivalent to $M(x,y)$$M(x,y)$]

      • Solve $N(x,y)+g^{\prime }(x)=M(x,y)$$N(x,y)+g^{\,\prime}(x)=M(x,y)$ for $g^{\prime }(x)$$g^{\,\prime}(x)$

      • ${\displaystyle g(x)=\int g^{\prime }(x)dx}$$\displaystyle g(x) = \int{g^{\,\prime}(x)\,dx}$

9. Thus, ${\displaystyle F(x,y)=\int M(x,y)dx+g(y)}$$\displaystyle F(x,y)=\int{M(x,y)}\,dx+g(y)$ OR ${\displaystyle F(x,y)=\int N(x,y)dy+g(x)}$$\displaystyle F(x,y)=\int{N(x,y)}\,dy+g(x)$ usually, written as $F(x,y)=C$$F(x,y)=C$

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