Modeling Situations with Differential Equations

Abigail Young

6 min read

Next Topic - Verifying Solutions for Differential Equations

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Study Guide Overview

This study guide covers modeling situations with differential equations. It explains what differential equations are and how they represent relationships between a function and its rate of change. It focuses on direct and inverse proportionality and provides examples of how to write differential equations based on verbal descriptions. Finally, it demonstrates how to model real-world scenarios using differential equations, including finding the constant of proportionality.

#7.1 Modeling Situations with Differential Equations

Welcome back to AP Calculus with Fiveable! We are now diving into one of the most valuable concepts in calculus to simulate real-life scenarios. Our focus in this section is modeling situations with differential equations. These equations involve rates of change and help us understand how a quantity changes with respect to another. Let’s get into it! ⬇️

#⛷️ Differential Equations

Differential equations involve derivatives and represent the relationship between a function and its rate of change. They help us understand how functions change with respect to the independent variable in real-world scenarios.

For example, a differential equation can look like $\frac{dy}{dx} = 5x$ . In this equation, $\frac{dy}{dx}$ represents the derivative of the function $y$ with respect to $x$ . The equation is saying that the rate of change of $y$ with respect to $x$ is equal to 5 times $x$ .

#🧠 Understanding Proportionality

Proportionality, which is the concept that two quantities vary in a consistent way with respect to each other, forms the basis of many differential equations! You can have two values be directly proportional to each other, or indirectly proportional to each other.

Directly Proportionality: If $a$ is proportional to $b$ , th...

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Question 1 of 12

What does $\frac{dy}{dx}$ represent in the differential equation $\frac{dy}{dx} = 2x$ ?

The rate of change of $x$ with respect to $y$

The derivative of $x$ with respect to $y$

The rate of change of $y$ with respect to $x$

The integral of $y$ with respect to $x$