Exact Equations

Definition

Exact equations are a type of first-order differential equation that can be expressed in the form $$M(x, y)dx + N(x, y)dy = 0$$, where the partial derivatives of functions $$M$$ and $$N$$ satisfy a specific condition. This condition states that the equation is exact if \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \). This relationship indicates that there exists a function whose total differential gives rise to the original equation, allowing for the solution to be found by integrating.

5 Must Know Facts For Your Next Test

1. To determine if a first-order differential equation is exact, check if \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \). If this equality holds, the equation is exact.
2. Exact equations can often be solved by finding a potential function $$F(x, y)$$ such that \( dF = M(x, y)dx + N(x, y)dy $$, leading to \( F(x, y) = C $$ for some constant $$ C $$.
3. If an equation is not exact, you can sometimes find an integrating factor that will make it exact, allowing you to solve it.
4. The solutions to exact equations represent curves in the plane, where each curve corresponds to a specific value of the constant $$ C $$ in the solution.
5. In practice, many real-world problems can be modeled using exact equations due to their ability to describe conservation laws or equilibrium states.

Review Questions

• How can you verify whether a given first-order differential equation is exact?
  • To verify if a first-order differential equation is exact, you need to identify the functions $$M(x, y)$$ and $$N(x, y)$$ from the equation in the form $$M(x, y)dx + N(x, y)dy = 0$$. Then calculate the partial derivative of $$M$$ with respect to $$y$$ and the partial derivative of $$N$$ with respect to $$x$$. If these two partial derivatives are equal, i.e., \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \), then the equation is exact and can be solved by finding an appropriate potential function.
• What steps would you take to solve an exact differential equation once you have determined it is exact?
  • Once you determine that a differential equation is exact, you proceed by finding a potential function $$F(x, y)$$ such that its total differential equals the given equation. This involves integrating the function $$M(x,y)$$ with respect to $$x$$ while treating $$y$$ as constant, yielding part of the potential function. Then you find the integration of $$N(x,y)$$ with respect to $$y$$ and combine results while ensuring any functions of one variable added during integration are correctly accounted for. Setting $$F(x,y) = C$$ gives you the implicit solution to the original differential equation.
• Discuss how exact equations relate to broader concepts in calculus and their application in real-world scenarios.
  • Exact equations play a crucial role in calculus as they provide systematic ways to solve certain types of first-order differential equations. They embody the concept of total differentials and illustrate how changes in multiple variables can be interrelated. In real-world scenarios, exact equations often arise in fields such as physics and engineering where conservation laws are prevalent. For instance, they can model systems in equilibrium or describe relationships between physical quantities like energy and momentum, leading to insights about system behavior and stability.

Related terms

Integrating Factor: A function that is multiplied with a non-exact differential equation to convert it into an exact equation.

Total Differential: The total change in a multivariable function, which can be expressed in terms of its partial derivatives.

Solution Curve: A curve that represents the set of all points that satisfy a given differential equation.