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

El método del factor integrante ofrece un enfoque general para resolver cualquier ecuación diferencial lineal de primer orden que no sea separable.El factor integrante se obtiene tomando la exponencial de la integral del coeficiente de y.Multiplicar ambos lados de la ecuación por este factor transforma el lado izquierdo en la derivada de un producto, permitiendo que se resuelva fácilmente.Por ejemplo, consideremos un coche que se mueve bajo una fuerza constante del motor mientras experimenta una resistencia del aire proporcional a su velocidad.Aplicar la segunda ley de Newton conduce a una ecuación diferencial lineal de primer orden que no es separable, la cual puede resolverse usando un factor integrante.El factor integrante se determina tomando la exponencial de la integral del coeficiente de velocidad respecto al tiempo, lo que ayuda a resolver la ecuación fácilmente.Multiplicar ambos lados por el factor integrante transforma el lado izquierdo en la derivada del producto del factor y la velocidad.Integrar ambos lados da una solución general que predice la velocidad del coche en cualquier momento.El resultado muestra que la velocidad cambia rápidamente al principio, según dicta el exponente negativo, y luego se acerca gradualmente a la velocidad terminal.

ecuaciones diferenciales lineales

Linear Differential Equations

El m&#xE9;todo del factor integrante proporciona una forma sistem&#xE1;tica de resolver ecuaciones diferenciales lineales de primer orden, especialmente aquellas que no pueden resolverse por separaci&#xF3;n de variables. Este m&#xE9;todo es particularmente &#xFA;til para modelar sistemas f&#xED;sicos dependientes del tiempo, influidos tanto por entradas constantes como por fuerzas resistivas. Un ejemplo com&#xFA;n es el movimiento de un autom&#xF3;vil sometido a una fuerza motriz constante del motor mientras experimenta una resistencia aerodin&#xE1;mica proporcional a su velocidad.
En estos escenarios, la segunda ley de Newton da lugar a una ecuaci&#xF3;n diferencial en la que la tasa de variaci&#xF3;n de la velocidad depende del equilibrio entre una fuerza motriz constante y una fuerza resistiva que aumenta con la velocidad. Dado que esta ecuaci&#xF3;n no es separable, se requiere el m&#xE9;todo del factor integrante. El paso clave consiste en multiplicar la ecuaci&#xF3;n por una funci&#xF3;n obtenida a partir del coeficiente del t&#xE9;rmino de la velocidad. Esta transformaci&#xF3;n reformula el lado izquierdo como la derivada de un producto, simplificando el proceso de integraci&#xF3;n y permitiendo obtener la soluci&#xF3;n directamente.
Al aplicarla a un autom&#xF3;vil que se mueve inicialmente a alta velocidad, la soluci&#xF3;n describe un proceso de desaceleraci&#xF3;n. Con el tiempo, la fuerza resistiva debida a la resistencia aerodin&#xE1;mica contrarresta progresivamente la fuerza del motor, lo que provoca una disminuci&#xF3;n gradual de la velocidad. La velocidad finalmente se estabiliza en un valor inferior y constante, conocido como velocidad terminal, donde las fuerzas se equilibran. La disminuci&#xF3;n exponencial de la soluci&#xF3;n caracteriza la rapidez con la que la velocidad pasa del valor inicial elevado a la velocidad terminal. El m&#xE9;todo del factor integrante no solo facilita esta soluci&#xF3;n, sino que tambi&#xE9;n proporciona una visi&#xF3;n de la din&#xE1;mica de sistemas donde la resistencia desempe&#xF1;a un papel crucial en el comportamiento a largo plazo.

Ecuaciones diferenciales lineales

El m&#xE9;todo del factor integrante proporciona una forma sistem&#xE1;tica de resolver ecuaciones diferenciales lineales de primer orden, especialmente ...

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Differential Equations

Ecuaciones diferenciales

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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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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

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

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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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