Exact equations example 1 | First order differential equations | Khan Academy

TL;DR

Exploring how to solve a given differential equation using the method of exact equations.

Transcript

OK, I filled your brain with a bunch of partial derivatives and psi's, with respect to x's and y's. I think now it's time to actually do it with a real differential equation, and make things a little bit more concrete. So let's say I have the differential, y, the differential equation, y cosine of x, plus 2xe to the y, plus sine of x, plus-- I'm al... Read More

Key Insights

• 🅰️ Exact equations are a type of differential equation where the partial derivatives of the equation satisfy a specific condition.
• 🏆 The test for exactness involves comparing the partial derivatives of the equation's functions.
• 🆘 Finding the potential function by integrating the equation's functions helps simplify the equation.
• ⏮️ Integration constants in the potential function can be determined by considering constants from previous steps or initial conditions.

Questions & Answers

Q: How can you determine if a differential equation is exact or not?

To test if an equation is exact, examine if the partial derivative of the function M with respect to y is equal to the partial derivative of the function N with respect to x. If they are equal, the equation is exact.

Q: What is the process for finding the potential function in an exact equation?

To find the potential function, integrate the function M with respect to x. The constant of integration might include a function of y due to taking the partial derivative. Similarly, integrate the function N with respect to y, and set it equal to the potential function.

Q: What is the significance of solving an exact equation?

Solving an exact equation allows for the determination of a potential function, which can then be used to rewrite the given differential equation in a simpler form. This simplification helps in finding the general solution to the equation.

Q: Is it possible for a differential equation to have multiple potential functions?

No, an exact equation can have only one potential function. The potential function can have a constant of integration, which depends on the given equation and initial conditions.

Summary & Key Takeaways

• The video discusses solving a specific differential equation using the method of exact equations.

• It explains the concept of exact equations and how to test if a given equation is exact.

• The process of finding the potential function and integrating to obtain the final solution is demonstrated.

• The video concludes by presenting the solution to the given differential equation.