Find the function f such that f ^ ' ( x ) = f ( x ) ( 1

Name: Problem 20, Chapter 9: Calculus Early Transcendentals
Uploaded: 2020-10-11T19:31:47-08:00
Duration: 3 min 58 s
Channel: Subham Jyoti Mishra
Description: Find the function f such that f ^ ' ( x ) = f ( x ) ( 1 - f ( x ) ) and f ( 0 ) = ( 1 )/( 2 ) .

step 1 So in this question we have the differential equation if dash x is equal to f of x to 1 minus f

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Key Concepts

Logistic Differential Equation

The logistic differential equation is a specific form of differential equation commonly used to model population growth with limiting factors. It includes a term representing growth and another term accounting for the saturation effect due to competition or resource limitations, leading to an S-shaped growth curve.

Initial Value Problem

An initial value problem in differential equations is one where the solution is required to satisfy a specific value, usually given at the start of the observation period, along with the differential equation. This additional condition allows the selection of a unique solution from a family of possible solutions.

Differential Equations

Differential equations involve functions and their derivatives, representing relationships that often model change in physical, biological, or economic systems. They are central in analyzing how systems evolve over time under various conditions.

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Separable differential equations are a class of differential equations in which the variables can be separated on different sides of the equation, allowing one to integrate each side with respect to its own variable. This technique simplifies solving by reducing a differential equation into two integrals, each involving only one variable.

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