Exact Equations

Definition

Exact equations are a special type of first-order linear differential equation where the equation can be integrated directly to obtain the general solution. These equations have a particular form that allows for a straightforward integration process, making them an important concept in the study of differential equations.

5 Must Know Facts For Your Next Test

1. Exact equations have the form $M(x,y)dx + N(x,y)dy = 0$, where $M$ and $N$ are functions of $x$ and $y$.
2. The condition for an equation to be exact is that the partial derivatives of $M$ and $N$ with respect to $y$ and $x$, respectively, are equal: $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.
3. To solve an exact equation, an integrating factor can be used to transform the equation into an exact form, allowing for direct integration to obtain the general solution.
4. Homogeneous first-order linear differential equations are a special case of exact equations where the coefficient of the dependent variable is a function of the independent variable only.
5. Separable first-order differential equations are a type of exact equation where the variables can be separated, leading to a straightforward integration process.

Review Questions

• Explain the defining characteristics of an exact equation and how they differ from other types of first-order linear differential equations.
  • Exact equations are a special type of first-order linear differential equation where the equation can be integrated directly to obtain the general solution. The defining characteristic of an exact equation is that the partial derivatives of the coefficients $M(x,y)$ and $N(x,y)$ with respect to $y$ and $x$, respectively, are equal: $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. This particular form allows for a straightforward integration process, unlike other types of first-order linear differential equations that may require more complex techniques, such as the use of an integrating factor or variable separation.
• Describe the role of the integrating factor in the solution of exact equations and explain how it transforms the equation into a form that can be integrated directly.
  • The integrating factor is a crucial component in the solution of exact equations. When a first-order linear differential equation is not in exact form, the integrating factor can be used to transform the equation into an exact equation. The integrating factor is a function that, when multiplied with the original equation, ensures that the resulting equation satisfies the condition for an exact equation: $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. This transformation allows for the direct integration of the exact equation to obtain the general solution, which would not be possible without the use of the integrating factor.
• Discuss the relationship between exact equations, homogeneous equations, and separable equations, and explain how they are interconnected within the context of first-order linear differential equations.
  • Exact equations, homogeneous equations, and separable equations are all related concepts within the context of first-order linear differential equations. Homogeneous equations are a special case of exact equations where the coefficient of the dependent variable is a function of the independent variable only. Separable equations are a type of exact equation where the variables can be separated, leading to a straightforward integration process. The interconnectedness of these concepts is that they all share the property of being solvable through direct integration, unlike other types of first-order linear differential equations that may require more advanced techniques. Understanding the relationships between these special cases of first-order linear differential equations is crucial for effectively solving and analyzing a wide range of problems in the study of differential equations.

Related terms

Integrating Factor:

An integrating factor is a function that, when multiplied with a first-order linear differential equation, transforms the equation into an exact equation, allowing for direct integration to find the general solution.

Homogeneous Equation: A homogeneous first-order linear differential equation is a special case of an exact equation where the coefficient of the dependent variable is a function of the independent variable only.

Separable Equation:

A separable first-order differential equation is a type of exact equation where the variables can be separated, allowing for integration of the equation in a straightforward manner.