Verifying solutions to differential equations | AP Calculus AB | Khan Academy

TL;DR

A differential equation has a solution in the form of a function, and not a single value. Different functions can be tested to see if they are solutions to the equation.

Transcript

• [Instructor] So let's write down a differential equation, the derivative of y with respect to x is equal to four y over x. And what we'll see in this video is the solution to a differential equation isn't a value or a set of values. It's a function or a set of functions. But before we go about actually trying to solve this or figure out all of th... Read More

Key Insights

• 😫 The solution to a differential equation is a function or a set of functions.
• 🏆 Functions can be tested to determine if they satisfy a given differential equation.
• ☺️ A function must satisfy the equation for all values of x within its domain to be considered a solution.
• ❓ Not all functions are solutions to differential equations.
• ❓ Differential equations can have multiple solutions.
• ❓ Different notations can be used to represent differential equations, but the process of testing solutions remains the same.

Questions & Answers

Q: How do you determine if a function is a solution to a differential equation?

To determine if a function is a solution to a differential equation, you need to find the derivative of the function and check if it satisfies the equation by substituting it in.

Q: Can any function be a solution to a differential equation?

No, not all functions are solutions to differential equations. The function must satisfy the equation for all values of x within its domain to be considered a solution.

Q: Is there only one solution to a differential equation?

No, there can be multiple solutions to a differential equation. A differential equation may have a general solution that includes a constant and can have an infinite number of specific solutions.

Q: How can you test if a function is a solution to a differential equation when using different notation?

To test if a function is a solution using different notation, you need to find the derivative of the function in the new notation and check if it satisfies the equation by substituting it in.

Summary & Key Takeaways

• The solution to a differential equation is a function or a set of functions, not a single value.

• Different functions can be tested to see if they satisfy the given differential equation.

• Functions that satisfy the differential equation are considered solutions.