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

A separable equation is a first-order differential equation that can be split into two independent parts&#8212;one containing only x and the other only y.The y terms are placed on one side and the x terms on the other, allowing separate integration.For example, consider a hot cup of tea cooling in a room.The rate at which the tea cools is proportional to the difference between its temperature and the room temperature. A negative sign is added to the proportionality constant k to indicate that the temperature decreases with time.The equation is separable because it can be rewritten with temperature terms on one side and time on the other. Integrating both sides gives a logarithmic equation with a constant of integration.When both sides are exponentiated, the general solution emerges, with a constant that may be positive, negative, or zero. The constant is found by substituting t equals zero and the initial temperature.The temperature difference decays exponentially, causing the tea to cool quickly at first and then more slowly over time.

separable differential equations

A separable differential equation is a type of first-order differential equation where the derivative dy/dx can be expressed as a product of two functions: one that depends only on x and another that depends only on y. This allows for the rearrangement of the equation so that all terms involving y are on one side, and all terms involving x are on the other. This process, known as the separation of variables, simplifies the process of solving the equation by enabling the integration of both sides independently.Integration After SeparationOnce the variables are separated, each side of the equation can be integrated with respect to its own variable. This integration yields a relationship between x and y, which may be implicit or explicit depending on the functions involved. For example, if a differential equation simplifies to expressions involving powers of x and y, integrating each side would lead to polynomial terms and a constant of integration.Determining the Particular SolutionTo obtain a specific solution, an initial condition, such as a known value of y at a particular x, is substituted into the general solution. This allows the constant of integration to be determined, yielding a unique solution that satisfies both the differential equation and the initial condition. This approach is especially useful in modeling real-world processes where initial values are known, such as in population growth, chemical reactions, or cooling laws.

Separable Differential Equations

A separable differential equation is a type of first-order differential equation where the derivative dy/dx can be expressed as a product of two ...

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Calculus

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Differential Equations

KEY TERMS AND CONCEPTS

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...

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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