Intro to Differential Equations, Modelling, Worksheet, Example - Calculus

A differential equation is an equation that involves an unknown function and its derivatives, describing how a quantity changes over time or space. These equations are fundamental to modeling real-world phenomena in fields like physics, biology, and engineering, where change is a central concept. The solution to a differential equation is not a single number but a function or set of functions that satisfy the equation's conditions.

💡What is a differential equation?
• An equation containing an unknown function (e.g., y = f(x)) and one or more of its derivatives.
• For example, dy/dx represents the rate of change of y with respect to x.

💡How are they used?
• Modeling real-world changes: They are used to describe processes like population growth, heat transfer, radioactive decay, and the motion of objects.
• A language of science: They provide a powerful mathematical framework for expressing the laws of physics and other scientific principles.

💡Key Concepts
• Function and Derivatives: The equation establishes a relationship between the function itself and its rate of change (its derivative).
• Solution is a function: Unlike algebraic equations, the solution to a differential equation is a function or a family of functions, not a specific value.
• Initial Conditions: To find a unique, "particular" solution, you need additional information called an initial condition (e.g., the starting position or velocity of an object).

💡Examples
• Newton's Law of Cooling: Describes how the temperature of an object changes over time as it cools down.
• Population Dynamics: Models how the size of a population changes based on factors like birth and death rates.
• Simple Harmonic Motion: Describes the oscillation of a mass attached to a spring, involving the second derivative of its position.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1DMK4EA0f8SfF4SgdiOZ39F73He_YWIwe/view?usp=drive_link
• Answers: https://drive.google.com/file/d/1QpzjiCPjfoxydzZqvjyJJsqv9jyRLZ0o/view?usp=drive_link

💡Chapters:
00:00 Intro to differential equations
01:00 Worked example

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