Name: solving separable differential equations
Uploaded: 2026-01-12T12:24:52.000Z
Duration: 1 min 20 s
Channel: JoVE

Une équation séparable est une équation différentielle du premier ordre qui peut être décomposée en deux parties indépendantes — l’une contenant seulement x et l’autre seulement y.Les termes y sont placés d’un côté et les termes x de l’autre, permettant une intégration séparée.Par exemple, imaginez une tasse de thé chaud refroidissant dans une pièce.La vitesse de refroidissement du thé est proportionnelle à la différence entre sa température et la température ambiante. Un signe négatif est ajouté à la constante de proportionnalité k pour indiquer que la température diminue avec le temps.L’équation est séparable car elle peut être réécrite avec des termes de température d’un côté et du temps de l’autre. En intégrant les deux côtés, on obtient une équation logarithmique avec une constante d’intégration.Lorsque les deux côtés sont exponentiés, la solution générale émerge, avec une constante pouvant être positive, négative ou nulle. La constante est trouvée en substituant t égal à zéro et en remplaçant la température initiale.La différence de température diminue de façon exponentielle, ce qui fait refroidir le thé rapidement au début puis plus lentement avec le temps.

Équations différentielles séparables

Une &#xE9;quation diff&#xE9;rentielle &#xE0; variables s&#xE9;parables est un type d&#x2019;&#xE9;quation diff&#xE9;rentielle du premier ordre o&#xF9; la d&#xE9;riv&#xE9;e dy/dx peut &#xEA;tre exprim&#xE9;e en produit de deux fonctions : l&#x2019;une d&#xE9;pendant uniquement de x et l&#x2019;autre uniquement de y. Cela permet de r&#xE9;arranger l&#x2019;&#xE9;quation de sorte que tous les termes contenant y se trouvent d&#x2019;un c&#xF4;t&#xE9; et tous les termes contenant x de l&#x2019;autre. Ce proc&#xE9;d&#xE9;, appel&#xE9; la s&#xE9;paration des variables, simplifie la r&#xE9;solution de l&#x2019;&#xE9;quation en permettant d&#x2019;int&#xE9;grer s&#xE9;par&#xE9;ment les deux membres.
Int&#xE9;gration apr&#xE8;s la s&#xE9;paration
Une fois les variables s&#xE9;par&#xE9;es, chaque membre de l&#x2019;&#xE9;quation peut &#xEA;tre int&#xE9;gr&#xE9; par rapport &#xE0; sa variable propre. Cette int&#xE9;gration &#xE9;tablit une relation entre x et y, qui peut &#xEA;tre implicite ou explicite selon les fonctions en jeu. Par exemple, si une &#xE9;quation diff&#xE9;rentielle se simplifie en expressions contenant des puissances de x et de y, l&#x2019;int&#xE9;gration de chaque membre conduira &#xE0; des termes polynomiaux et &#xE0; une constante d&#x2019;int&#xE9;gration.
D&#xE9;termination de la solution particuli&#xE8;re
Pour obtenir une solution particuli&#xE8;re, une condition initiale, telle qu&#x2019;une valeur connue de y pour une valeur particuli&#xE8;re de x, est substitu&#xE9;e dans la solution g&#xE9;n&#xE9;rale. Cela permet de d&#xE9;terminer la constante d&#x2019;int&#xE9;gration et conduit &#xE0; une solution unique qui satisfait &#xE0; la fois l&#x2019;&#xE9;quation diff&#xE9;rentielle et la condition initiale. Cette approche est particuli&#xE8;rement utile pour mod&#xE9;liser des ph&#xE9;nom&#xE8;nes r&#xE9;els o&#xF9; les valeurs initiales sont connues, comme la croissance d&#xE9;mographique, les r&#xE9;actions chimiques ou les lois de refroidissement.

Une &#xE9;quation diff&#xE9;rentielle &#xE0; variables s&#xE9;parables est un type d&#x2019;&#xE9;quation diff&#xE9;rentielle du premier ordre o&#xF9; la ...

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Calculus

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Differential Equations

Équations différentielles

KEY TERMS AND CONCEPTS

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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 implicit or explicit.

If the equation includes powers of x and y, integrating each side can produce polynomial terms and a constant of integration. To find a particular solution, an initial condition is used, such as a known value of y at a specific x. Substituting that value lets you solve for the constant and get a unique solution.

This method is useful in models where an initial value is known. It appears in real-world situations such as population growth, chemical reactions, and cooling laws.

Solving Separable Differential Equations

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...

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

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 to separate variables, integrate both sides, and use an initial condition to find a particular solution for separable differential equations.

Solving Separable Differential 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

Solving Separable Differential Equations

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