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

A differential equation is an equation that contains an unknown function and one or more of its derivatives. The order of a differential equation is given by the highest order derivative present in the equation.When the equation involves only the first derivative, it is called first order. If it involves the second derivative, it is called second order, and so on.For example, a spring-mass system can be modeled as a simple harmonic oscillator, which equates the net force on the mass and the restoring force in the spring, described by Hooke&#8217;s law for linear springs.The spring force acts in the direction opposite to the displacement and is proportional to how far the mass moves from its equilibrium position.Newton&#8217;s second law links the net force on the mass to its acceleration. Combining these two relationships gives an equation in which mass times acceleration equals a negative constant times displacement.By rewriting acceleration as the second derivative of displacement over time, a second-order differential equation is obtained.This second derivative links the motion of the system to a differential equation.

introduction to differential equations

A differential equation is a mathematical expression that establishes a relationship between a function and its derivatives. These equations are fundamental in modeling dynamic systems across various fields of science and engineering. The order of a differential equation is defined by the highest order derivative present in the equation. A first-order differential equation includes only the first derivative, while a second-order differential equation includes up to the second derivative of the unknown function.Second-Order Differential Equations in Physical SystemsA classic example of a second-order differential equation arises from modeling the motion of a spring&#8211;mass system. In this idealized system, a mass is attached to a spring that obeys Hooke&#8217;s Law and is free from any damping or external forces. When the mass is displaced from its equilibrium position, the spring exerts a restoring force that is linearly proportional to the displacement and acts in the opposite direction. Mathematically, this restoring force is given by<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:8.91%;"><img style="aspect-ratio:78/32;" src="https://cloudfront.jove.com/files/ftp_upload/40597/40597_001_editor_40597.svg" alt="Hooke's Law formula, Fs=-kx, spring force relation, physics, elasticity concept." data-altupdatedat="1775223722000" data-alt="40597_001.svg" /></figure>where k is the spring constant and x is the displacement from equilibrium.According to Newton&#8217;s second law, the net force acting on an object is equal to the product of its mass and acceleration. Substituting the spring force into Newton&#8217;s law yields<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:11.77%;"><img style="aspect-ratio:81/44;" src="https://cloudfront.jove.com/files/ftp_upload/40597/40597_002_editor_40597.svg" alt="Harmonic oscillator equation m(d&#178;x/dt&#178;)=-kx, describing simple harmonic motion, physics formula." data-altupdatedat="1775223722000" data-alt="40597_002.svg" /></figure>When acceleration is expressed as the second derivative of displacement with respect to time.Rearranging terms leads to the standard form of the second-order linear differential equation:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:15%;"><img style="aspect-ratio:148/47;" src="https://cloudfront.jove.com/files/ftp_upload/40597/40597_003_editor_40597.svg" alt="Harmonic oscillator equation: \(m \frac{{d^2x}}{{dt^2}} + kx = 0\); physics dynamics formula." data-altupdatedat="1775223722000" data-alt="40597_003.svg" /></figure>This equation describes simple harmonic motion and serves as a foundational model in the study of mechanical vibrations and oscillatory systems.

Introduction to Differential Equations

A differential equation is a mathematical expression that establishes a relationship between a function and its derivatives. These equations are ...

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Differential Equations

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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

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

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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

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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...

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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

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