Solve the given differential equation by separation of variables. (d X)/(d t)=(10-X)(50-X)

Name: Problem 18, Chapter 8: Calculus, Early Transcendentals
Uploaded: 2021-08-15T06:25:25-08:00
Duration: 2 min 51 s
Channel: Darshan Maheshwari
Description: Solve the given differential equation by separation of variables. (d X)/(d t)=(10-X)(50-X)

step 1 let's talk about this question we need to solve this differential equation by separation of vari

Solve the given differential equation by separation of...

Key Concepts

Partial Fractions Decomposition

Partial fractions decomposition is a method used to simplify the integration of rational functions. When the separated differential equation contains a rational expression with factorable denominators, this technique breaks it into simpler fractions that are easier to integrate, facilitating the solution process.

Integration

Integration is the fundamental technique used after separating the variables. By integrating both sides of a separated equation, one can accumulate the contributions of the changes described by the differential equation, resulting in an implicit or explicit general solution.

Integration Constant

An integration constant arises when indefinite integrals are computed during the solution of a differential equation. It represents the family of solutions incorporating initial conditions or boundary values, and is crucial for determining the particular solution in real-world scenarios.

Differential Equations

Differential equations involve functions and their derivatives, encapsulating relationships between rates of change and the quantities themselves. They are essential in modeling a variety of phenomena in physics, biology, economics, and engineering by expressing how a system evolves over time or space.

Separation of Variables

Separation of variables is a method used to solve ordinary differential equations, whereby the equation is manipulated so that all terms involving the dependent variable are on one side and all terms involving the independent variable are on the other. This allows the integration of each side with respect to its own variable, leading to a solution.

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