Understanding Exact Differential Equations

The video tutorial explains how to solve a given differential equation by verifying its exactness. It involves checking the partial derivatives of the functions M and N, ensuring they are equal. Once confirmed, the tutorial demonstrates integrating the equation to find the potential function and derive the general solution. The process includes integrating with respect to x and y, and determining the function of y. The final solution is presented as an implicit equation.

What is the first step in solving the given differential equation?

• Check if it is linear

• Check if it is exact

• Check if it is homogeneous

• Check if it is separable

What are the expressions for M(x, y) and N(x, y) in the given differential equation?

• M(x, y) = x^2 + y, N(x, y) = x^2 - 4y

• M(x, y) = 2x - y, N(x, y) = x + 4y

• M(x, y) = 2x + y, N(x, y) = x - 4y

• M(x, y) = x + y, N(x, y) = x - y

What is the result of differentiating M(x, y) with respect to y?

• 0

• 2

• 1

• -1

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

• 2x^2 + y + C

• x^2 + y^2 + C

• x^2 + x + C

• x^2 + xy + C

What does the function of Y represent in the integration of the first equation?

• A constant

• A function of x

• A variable

• A function of y

What is the derivative of a function of Y with respect to Y?

• A prime of Y

• A constant

• Zero

• A function of x

What is the integral of -4y with respect to y?

• -2y^2

• 4y^2

• -4y^2

• 2y^2

What is the potential function F(x, y) after determining the function of Y?

• x^2 - xy - 2y^2

• x^2 + xy - 2y^2

• x^2 + xy + 2y^2

• x^2 - xy + 2y^2

What is the general solution of the differential equation?

• x^2 + xy + 2y^2 = C

• x^2 + xy - 2y^2 = C

• x^2 - xy + 2y^2 = C

• x^2 - xy - 2y^2 = C

What is the form of the solution to the exact differential equation?

• F(x, y) = C

• F(x, y) = xy

• F(x, y) = 0

• F(x, y) = x + y