Name: integrating factors for motion with drag
Uploaded: 2026-01-12T12:30:56.000Z
Duration: 1 min 27 s
Channel: JoVE

The integrating factor method gives a general approach for solving any first-order linear differential equation that is not separable.The integrating factor is found by taking the exponential of the integral of the coefficient of y.Multiplying both sides of the equation by this factor transforms the left-hand side into the derivative of a product, allowing it to be easily solved.For example, consider a car moving under a constant engine force while experiencing air resistance proportional to its speed.Applying Newton&#8217;s second law leads to a first-order linear differential equation that is not separable, which can be solved using an integrating factor.The integrating factor is found by taking the exponential of the integral of the velocity's coefficient with respect to time, which helps solve the equation easily.Multiplying both sides by the integrating factor transforms the left-hand side into the derivative of the product of the factor and the speed.Integrating both sides gives a general solution that predicts the car&#8217;s speed at any time.The result shows that speed changes quickly at first, as dictated by the negative exponent, then gradually approaches the terminal velocity.

linear differential equations

The integrating factor method provides a systematic way to solve first-order linear differential equations, especially those that cannot be handled by separation of variables. This method is particularly useful in modeling time-dependent physical systems influenced by both constant inputs and resistive forces. A common example is the motion of a car subjected to a constant engine force while experiencing air resistance proportional to its velocity.In such scenarios, Newton&#8217;s second law yields a differential equation where the rate of change of velocity depends on a balance between a constant driving force and a resistive force that increases with speed. Since this equation is not separable, the integrating factor method is required. The key step involves multiplying the equation by a function derived from the coefficient of the velocity term. This transformation recasts the left-hand side as the derivative of a product, simplifying the integration process and allowing the solution to be obtained directly.When applied to a car initially moving at a high speed, the solution describes a deceleration process. Over time, the resistive force due to air drag increasingly counteracts the engine force, leading to a gradual reduction in speed. The velocity eventually stabilizes at a lower, constant value known as the terminal velocity, where the forces are balanced. The exponential decay in the solution characterizes how quickly the speed transitions from the initial high value to the terminal velocity. The integrating factor method not only facilitates this solution but also provides insight into the dynamics of systems where resistance plays a critical role in long-term behavior.

Linear Differential Equations

The integrating factor method provides a systematic way to solve first-order linear differential equations, especially those that cannot be handled by ...

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JoVE Core

Calculus

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Differential Equations

KEY TERMS AND CONCEPTS

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 law gives a differential equation for velocity, and the rate of change of velocity depends on both the engine force and drag proportional to speed. Because this equation is not separable, the integrating factor method is needed to solve it.

The key step is multiplying the equation by a function built from the coefficient of the velocity term. This turns the left-hand side into the derivative of a product, which makes integration straightforward. After solving, the velocity of a fast-moving car drops over time.

The solution shows deceleration toward a lower steady speed. That limiting speed is the terminal velocity, where the driving force and air resistance balance each other. The exponential form of the solution shows how quickly the motion approaches that final value and highlights how resistance shapes the long-term behavior of the system.

Integrating Factors for Motion with Drag

introduction to differential equations

Modeling Motion with Second-Order Equations

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modeling with differential equations

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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...

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 the integrating factor method solves first-order linear differential equations using a car model with air resistance, deceleration, and terminal velocity.

Integrating Factors for Motion with Drag | 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

Integrating Factors for Motion with Drag

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