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

Równanie separowalne to równanie różniczkowe pierwszego rzędu, które można podzielić na dwie niezależne części — jedną zawierającą tylko x , a drugą tylko y.Wyrazy y są umieszczone po jednej stronie, a wyrazy x po drugiej, co pozwala na oddzielną całkowanie.Na przykład rozważmy gorącą filiżankę herbaty chłodzącą w pomieszczeniu.Szybkość stygnięcia herbaty jest proporcjonalna do różnicy między jej temperaturą a temperaturą pokojową. Do stałej proporcjonalności k dodaje się znak ujemny, aby wskazać, że temperatura spada z czasem.Równanie jest separowalne, ponieważ można je przepisać z temperaturą po jednej stronie i czasem z drugiej. Całkowanie obu stron daje równanie logarytmiczne o stałej całkowania.Gdy obie strony są wykładnicze, powstaje ogólne rozwiązanie, z stałą, która może być dodatnia, ujemna lub zerowa. Stałą otrzymuje się, podstawiając t równe zero i temperaturze początkowej.Różnica temperatur zmniejsza się wykładniczo, powodując, że herbata najpierw szybko stygnie, a z czasem coraz wolniej.

równania różniczkowe rozdzielne

R&#xF3;wnanie r&#xF3;&#x17C;niczkowe rozdzielne to rodzaj r&#xF3;wnania r&#xF3;&#x17C;niczkowego pierwszego rz&#x119;du, w kt&#xF3;rym pochodn&#x105; dy/dx mo&#x17C;na przedstawi&#x107; jako iloczyn dw&#xF3;ch funkcji, z kt&#xF3;rych jedna zale&#x17C;y wy&#x142;&#x105;cznie od x, a druga wy&#x142;&#x105;cznie od y. Pozwala to na przekszta&#x142;cenie r&#xF3;wnania r&#xF3;&#x17C;niczkowego tak, aby wszystkie wyrazy z y znajdowa&#x142;y si&#x119; po jednej stronie, a wszystkie wyrazy z x po drugiej. Ten proces, znany jako rozdzielanie zmiennych, upraszcza rozwi&#x105;zywanie r&#xF3;wnania r&#xF3;&#x17C;niczkowego, umo&#x17C;liwiaj&#x105;c ca&#x142;kowanie obu stron niezale&#x17C;nie.
Ca&#x142;kowanie po rozdzieleniu
Po rozdzieleniu zmiennych ka&#x17C;d&#x105; stron&#x119; r&#xF3;wnania mo&#x17C;na ca&#x142;kowa&#x107; wzgl&#x119;dem jej w&#x142;asnej zmiennej. To ca&#x142;kowanie prowadzi do zale&#x17C;no&#x15B;ci mi&#x119;dzy x i y, kt&#xF3;ra mo&#x17C;e by&#x107; jawna lub niejawna, w zale&#x17C;no&#x15B;ci od postaci funkcji. Na przyk&#x142;ad, je&#x15B;li r&#xF3;wnanie r&#xF3;&#x17C;niczkowe sprowadza si&#x119; do wyra&#x17C;e&#x144; z pot&#x119;gami x i y, ca&#x142;kowanie ka&#x17C;dej strony prowadzi do wielomian&#xF3;w i sta&#x142;ej ca&#x142;kowania.
Wyznaczanie rozwi&#x105;zania szczeg&#xF3;lnego
Aby uzyska&#x107; rozwi&#x105;zanie szczeg&#xF3;lne, do rozwi&#x105;zania og&#xF3;lnego podstawia si&#x119; warunek pocz&#x105;tkowy, taki jak znana warto&#x15B;&#x107; y dla okre&#x15B;lonego x. Pozwala to wyznaczy&#x107; sta&#x142;&#x105; ca&#x142;kowania, co daje jednoznaczne rozwi&#x105;zanie spe&#x142;niaj&#x105;ce zar&#xF3;wno r&#xF3;wnanie r&#xF3;&#x17C;niczkowe, jak i warunek pocz&#x105;tkowy. To podej&#x15B;cie jest szczeg&#xF3;lnie przydatne w modelowaniu rzeczywistych proces&#xF3;w, takich jak wzrost populacji, reakcje chemiczne, prawa ch&#x142;odzenia oraz inne procesy fizyczne, w kt&#xF3;rych znane s&#x105; warto&#x15B;ci pocz&#x105;tkowe.

Równania różniczkowe rozdzielne

R&#xF3;wnanie r&#xF3;&#x17C;niczkowe rozdzielne to rodzaj r&#xF3;wnania r&#xF3;&#x17C;niczkowego pierwszego rz&#x119;du, w kt&#xF3;rym pochodn&#x105; ...

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Differential Equations

Równania różniczkowe

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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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