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➗calculus ii review

First-Order Equations

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

First-order equations are a type of differential equation where the highest derivative present in the equation is the first derivative. These equations describe the relationship between a function and its first derivative, and they are fundamental in the study of differential equations and their applications.

5 Must Know Facts For Your Next Test

  1. First-order equations are the simplest and most fundamental type of differential equations, and they often arise in various fields, such as physics, chemistry, biology, and engineering.
  2. The general form of a first-order differential equation is $\frac{dy}{dx} = f(x, y)$, where $f(x, y)$ is a function of the independent variable $x$ and the dependent variable $y$.
  3. Solving a first-order differential equation typically involves finding a function $y(x)$ that satisfies the equation, which can be done using various techniques, such as separation of variables, substitution, or integrating factors.
  4. The solution to a first-order differential equation is often a function that represents the relationship between the independent and dependent variables, and it can be used to make predictions or analyze the behavior of the system being modeled.
  5. First-order differential equations are essential in the study of many physical and mathematical phenomena, including radioactive decay, population growth, and the motion of objects under the influence of forces.

Review Questions

  • Explain the general form of a first-order differential equation and how it differs from higher-order differential equations.
    • The general form of a first-order differential equation is $\frac{dy}{dx} = f(x, y)$, where $f(x, y)$ is a function of the independent variable $x$ and the dependent variable $y$. This form differs from higher-order differential equations, which involve higher derivatives, such as the second derivative $\frac{d^2y}{dx^2}$ or the third derivative $\frac{d^3y}{dx^3}$. The key distinction is that first-order equations only depend on the first derivative, making them simpler to analyze and solve compared to higher-order equations.
  • Describe the various techniques that can be used to solve first-order differential equations, and explain the conditions under which each technique is applicable.
    • There are several techniques used to solve first-order differential equations, including separation of variables, substitution methods (such as for homogeneous equations), and integrating factors. Separation of variables is applicable when the equation can be written in the form $\frac{dy}{dx} = f(x)g(y)$, allowing the variables to be separated and solved by integration. Substitution methods, like the one used for homogeneous equations, involve a change of variable to transform the equation into a separable form. Integrating factors are used for exact equations, where the equation can be written as a total derivative. The choice of technique depends on the specific form of the first-order differential equation and the conditions it satisfies.
  • Discuss the importance of first-order differential equations in modeling real-world phenomena and explain how their solutions can be used to make predictions or analyze the behavior of the system being modeled.
    • First-order differential equations are fundamental in the study of many physical, chemical, biological, and engineering systems, as they can be used to model a wide range of phenomena. For example, first-order equations are used to model radioactive decay, population growth, the motion of objects under the influence of forces, and the flow of fluids. The solutions to these equations, which represent the relationship between the independent and dependent variables, can be used to make predictions about the behavior of the system, such as the rate of decay, the growth of a population, or the trajectory of a moving object. By analyzing the properties of the solutions, such as their equilibrium points, stability, and transient behavior, researchers and engineers can gain valuable insights into the dynamics of the system being modeled and use this information to inform decision-making and design processes.