Understanding Exact Differential Equations

The video tutorial explains how to solve a given differential equation by checking for exactness and verifying it as an exact differential equation. It involves calculating partial derivatives of functions M and N, integrating to find the potential function F, and determining the constant function a(Y). The tutorial concludes with formulating the general solution and providing final remarks.

What is the first step in verifying if a differential equation is exact?

• Integrate M with respect to X.

• Ensure the partial derivative of M with respect to Y equals the partial derivative of N with respect to X.

• Differentiate N with respect to Y.

• Check if M and N are equal.

What is the partial derivative of cosine X with respect to Y?

• cosine X

• 0

• 1

• sine X

Which rule is applied to differentiate y e^(xy) with respect to Y?

• Chain rule

• Product rule

• Quotient rule

• Power rule

What is the result of differentiating x e^(xy) with respect to X?

• e^(xy)

• x y e^(xy) + e^(xy)

• y e^(xy)

• x e^(xy)

What does the equality of partial derivatives indicate about the differential equation?

• It is separable.

• It is linear.

• It is exact.

• It is not exact.

What is the integral of cosine X with respect to X?

• cosine X

• secant X

• sine X

• tangent X

Tags

CCSS.8.EE.C.8B

What substitution is used to integrate y e^(xy) with respect to X?

• U = x

• U = y

• U = xy

• U = e^(xy)

Tags

CCSS.8.F.B.4

CCSS.HSF.LE.A.2

What must a'(Y) equal for the function a(Y) to be a constant?

• 0

• Y

• X

• 1

Tags

CCSS.8.EE.C.8B

What is the general solution of the differential equation?

• sine X + e^(xy) = Y

• sine X + e^(xy) = c

• sine X + e^(xy) = 0

• cosine X + e^(xy) = c

What is the role of the constant C in the general solution?

• It is an arbitrary constant representing the family of solutions.

• It represents a specific solution.

• It is the derivative of the solution.

• It is the integral of the solution.