Exact differential equation (introduction & example)

TL;DR

Learn how to solve differential equations using implicit differentiation and the exact equation method.

Transcript

in this video I will go over the idea you can decide questions however before I talk about this let me ask you guys for review purpose what it's a differential equation well simple right is just an equation that has a derivative and now let's think back back to calc 1 how do we get the derivative well ideally speaking you wonder why to be isolated ... Read More

Key Insights

• 👻 Implicit differentiation allows us to find derivatives when the dependent variable is not isolated.
• ❓ The total differential formula involves partial derivatives and is used in solving exact equations.
• ❓ Exact equations involve finding a function that satisfies the equation and using it to solve the equation.
• ❓ The process of solving exact equations involves finding partial derivatives, integrating, and solving for constants.
• ❓ The derivative obtained through implicit differentiation can be used to solve differential equations.
• ❓ The exact equation method provides an alternative approach to solving differential equations.
• ✅ Checking for exactness is necessary before applying the exact equation method.

Questions & Answers

Q: What is differential equation and how do we find its derivative using implicit differentiation?

A differential equation is an equation that involves a derivative. To find its derivative using implicit differentiation, we differentiate the equation, treating the dependent variable as a function of the independent variable.

Q: How do we solve a differential equation when the dependent variable is not isolated?

When the dependent variable is not isolated, we can still find the derivative using implicit differentiation. We differentiate each term, using the product and chain rules when necessary, and solve for the derivative.

Q: What is the total differential formula and how does it relate to exact equations?

The total differential formula involves partial derivatives and is used to solve exact equations. It represents the total change in a function due to changes in both variables. In exact equations, we equate the partial derivatives to find the function and use it to solve the equation.

Q: How does the process of solving exact equations work?

To solve an exact equation, we start by finding the partial derivative with respect to one variable and integrate it to find the function. Then, we find the partial derivative with respect to the other variable and equate it to the derivative in the equation. Finally, we solve for the constants and simplify the equation.

Summary & Key Takeaways

• Implicit differentiation allows us to find the derivative of an equation when the dependent variable cannot be isolated.

• The total differential formula involves partial derivatives and can be used to solve exact equations.

• In the exact equation method, we start by finding the partial derivative with respect to one variable and integrate to find the function. We then find the partial derivative with respect to the other variable and equate it to the derivative in the equation.