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Verifying Solutions for Differential Equations

Samuel Baker

Samuel Baker

6 min read

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Study Guide Overview

This study guide covers verifying solutions for differential equations. It explains the process of verifying solutions by substituting the derivative of a proposed solution into the differential equation. It includes examples demonstrating the verification process with step-by-step solutions. Key terms covered are general solutions and the verification process itself, using the product rule and differentiation techniques. Practice problems are also provided to reinforce the concepts.

7.2 Verifying Solutions for Differential Equations

In AP Calculus, one of the fascinating things we learn is how to solve differential equations. In this section, we will focus on verifying solutions to differential equations, a critical skill in both mathematics and real-world problem solving.

✅ Verifying Solutions

While actually solving a differential equation may seem daunting, verifying a given solution is a piece of cake! 🍰

Differential equations often have not just one, but infinitely many solutions. These solutions are known as general solutions. Each of these solutions can be tweaked slightly by adding different constants, and yet, they still solve the original differential equation. Imagine a family of curves on a graph, each differing slightly from the others, but all fitting the same overall pattern described by the differential equation.

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Example of a family of solutions from the general solution ax^b + cy^d.

Image courtesy of Wolfram

This verification process is rooted in understanding derivatives and how they function. When you're given a differential equation and a potential solution, your job is to take the derivative of the proposed solution and see if it fits perfectly into the original equation. It's like havin...

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Question 1 of 9

Which of the following represents a general solution to a differential equation?

y=x2y = x^2

y=x3+Cy = x^3 + C

y=sin(x)y = sin(x)

y=e2xy=e^{2x}