Exact Differential Equations Concepts

This video tutorial explains exact first-order differential equations, focusing on identifying and solving them. It begins with an introduction to the concept and properties of exact differential equations. The tutorial then describes how to test for exactness using partial derivatives. An example problem is presented, demonstrating the process of solving an exact differential equation. Finally, the video outlines the steps to find the solution, including integration and verifying the solution graphically.

What is the form of an exact first order differential equation?

• M(x, y) dy + N(x, y) dx = 0

• M(x, y) dx + N(x, y) dy = 0

• M(x) dx + N(y) dy = 0

• M(y) dy + N(x) dx = 0

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CCSS.HSF.IF.A.1

CCSS.8.F.A.1

What must be true for a differential equation to be exact?

• M and N must be linear functions

• M and N must be constants

• Partial derivative of M with respect to y equals partial derivative of N with respect to x

• Partial derivative of M with respect to x equals partial derivative of N with respect to y

Tags

CCSS.HSF.IF.A.1

CCSS.8.F.A.1

In the example, what is the function M(x, y)?

• x^2 + y

• 2xy

• 2x + y

• x^2 - 1

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CCSS.HSF.LE.A.2

CCSS.8.F.B.4

What is the result of the test for exactness in the example?

• The equation is linear

• The equation is exact

• The equation is homogeneous

• The equation is not exact

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CCSS.HSA-REI.B.4B

What is the integral of M(x, y) with respect to x in the example?

• x^2 y + h(y)

• x^2 + y

• 2xy + h(y)

• 2x^2 y + h(y)

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CCSS.HSF.LE.A.2

CCSS.8.F.B.4

What is the function h(y) in the example?

• h(y) = x

• h(y) = y

• h(y) = 0

• h(y) = x^2

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CCSS.8.EE.C.8B

What is the purpose of finding the function f(x, y) in solving exact differential equations?

• To determine the slope of the curve

• To find the general solution

• To calculate the area under the curve

• To identify the critical points

Tags

CCSS.8.EE.C.8B

What is the general solution form for the exact differential equation in the example?

• 2xy - y = c

• 2x^2 y + y = c

• x^2 y - y = c

• x^2 y + y = c

How can the solutions to the exact differential equation be visualized?

• By integrating N(x, y)

• By solving for y in terms of x

• By graphing the solutions for different values of c

• By plotting the function M(x, y)

What does the constant C represent in the solution of an exact differential equation?

• The y-intercept

• The slope of the tangent

• A family of solutions

• A specific solution