Name: modeling motion with second-order equations
Uploaded: 2026-01-12T12:25:00.000Z
Duration: 1 min 20 s
Channel: JoVE

微分方程是指包含一个未知函数及其一个或多个导数的方程。微分方程的阶数由方程中出现的最高阶导数决定。.当方程仅包含一阶导数时，称为一阶方程；若包含二阶导数，则称为二阶方程，依此类推。例如，弹簧-质量系统可以被建模为简谐振子，该模型将作用在质量上的合力与弹簧中的恢复力等同起来，而恢复力由胡克定律对线性弹簧进行描述。弹簧力的方向与位移方向相反，且与物体离开平衡位置的距离成正比。牛顿第二定律将物体所受的合外力与其加速度联系起来。结合这两个关系可得到一个方程，其中质量乘以加速度等于一个负常数乘以位移。通过将加速度重写为位移对时间的二阶导数，可得到一个二阶微分方程。这一二阶导数将系统的运动与一个微分方程联系起来。

微分方程导论

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Differential Equations

微分方程

KEY TERMS AND CONCEPTS

关键术语与概念

Introduction to Differential Equations

modeling motion with second-order equations

Differential equations connect a function with its derivatives. They are used to describe changing systems in science and engineering. A differential equation is classified by order, which is the highest derivative it contains. A first-order equation includes only the first derivative, while a second-order equation includes the second derivative of the unknown function.

A common second-order differential equation comes from a spring-mass system. In this idealized model, a mass is attached to a spring that follows Hooke’s Law and has no damping or external forces. When the mass moves away from equilibrium, the spring pulls it back. The restoring force is proportional to the displacement and points in the opposite direction.

Newton’s second law links that force to motion. It states that net force equals mass times acceleration. When the spring force is substituted into Newton’s law, and acceleration is written as the second derivative of displacement with respect to time, the result is the standard second-order linear differential equation for the system. This equation describes simple harmonic motion and is a key model for mechanical vibrations and oscillating systems.

Modeling Motion with Second-Order Equations

modeling-motion-with-second-order-equations

introduction to differential equations

Differential equations connect a function with its derivatives. They are used to describe changing systems in science and engineering. A differential equation is classified by order, which is the highest derivative it contains. A first-order equation includes only the first derivative, while a second-order equation includes the second derivative of the unknown function. A common second-order differential equation comes from a spring-mass system. In this idealized model, a mass is attached to a...

modeling with differential equations

Population Growth: Exponential vs. Logistic

Population growth can be modeled with differential equations by treating population size, P(t), as a function of time. The derivative of P(t) shows how fast the population changes. A simple starting idea is that the growth rate is proportional to the population size. That assumption gives an exponential growth model. In this model, the population rises continuously and faster over time. It can be a useful first approximation, but it does not match long-term behavior in nature. Real...

separable differential equations

Solving Separable Differential Equations

A separable differential equation is a first-order differential equation that can be written with dy/dx as a product of one function of x and one function of y. This form makes it possible to separate the variables so that all y terms stay on one side and all x terms stay on the other. After the variables are separated, each side can be integrated with respect to its own variable. The result is a relationship between x and y. Depending on the functions involved, that relationship may be...

orthogonal trajectories

Perpendicular Curves in Calculus

Orthogonal trajectories are a pair of curve families that cross at right angles. In this example, the starting family is made of parabolas that open sideways along the x-axis. They share the same shape, but a scaling parameter changes how wide each curve is. Every curve in the family passes through the origin and spreads out at a different rate. To find the orthogonal trajectories, the slope of each parabola must be calculated first. The slope of a tangent shows how steep the curve is at a...

linear differential equations

Integrating Factors for Motion with Drag

The integrating factor method solves first-order linear differential equations that are not separable. It is especially useful when a system has a constant driving force and a resistive force that depends on the changing variable. In these problems, the equation can be rearranged so the left-hand side becomes a product rule after multiplication by an integrating factor. A clear example is a car moving under a constant engine force while air resistance acts against its motion. Newton’s second...

differential equations: problem solving

Differential Equations and Falling Motion

Differential equations can model the motion of a falling object when gravity and air resistance act at the same time. In this example, a heavy test weight is released from rest during a ship safety check. Gravity pulls the weight downward, and air resistance pushes upward as the speed increases. Newton’s Second Law links the net force on the weight to its acceleration. Gravity provides a constant downward force equal to mass times gravitational acceleration. Air resistance is modeled as a...

key terms and concepts

Learn how differential equations describe changing systems and how a spring-mass model leads to a second-order equation for simple harmonic motion.

Modeling Motion with Second-Order Equations | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Modeling Motion with Second-Order Equations

Description

Transcript