Exact equations example 2 | First order differential equations | Khan Academy

TL;DR

Examples of solving exact differential equations using the methodology of finding a function psi.

Transcript

Let's do some more examples with exact differential equations. And I'm getting these problems from page 80 of my old college differential equations books. This is the fifth edition of Elementary Differential Equations by William Boyce and Richard DiPrima. I want to make sure they get credit, that I'm not making up these problems. I'm getting it fro... Read More

Key Insights

• 🙃 The methodology for solving exact differential equations involves finding a function psi that satisfies certain partial derivative conditions.
• 🙃 If a differential equation is exact, the function psi can be used to rewrite the equation in a simpler form.
• 🙃 It is possible to verify if the solution obtained from the methodology is correct by taking the derivative of psi and comparing it to the original differential equation.

Questions & Answers

Q: How can you determine if a given differential equation is exact?

To determine if a differential equation is exact, you need to check if the partial derivative of M with respect to y is equal to the partial derivative of N with respect to x. If they are equal, the equation is exact.

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

An exact differential equation allows us to find a function psi, where the partial derivative of psi with respect to x is equal to M and the partial derivative of psi with respect to y is equal to N. This enables us to rewrite the equation as dx/dx = 0.

Q: Can a differential equation be solved using the exact methodology if it is not an exact differential equation?

No, if a differential equation is not exact, we cannot solve it using the exact methodology. We need to employ other methods to solve non-exact differential equations.

Q: What is the solution to an exact differential equation?

The solution to an exact differential equation can be found by integrating the function psi with respect to x and setting it equal to a constant, c. The solution equation will then be psi = c.

Summary & Key Takeaways

• The video provides examples of solving exact differential equations using the methodology of finding a function psi.

• The first example given is an exact differential equation, which is determined by checking if the partial derivative of M with respect to y equals the partial derivative of N with respect to x.

• The second example given is not an exact differential equation because the partial derivative of M with respect to y is not equal to the partial derivative of N with respect to x.