Differential Equations and Integrals

This video tutorial explains how to solve an exact first-order differential equation. It begins by introducing the form of exact differential equations and the conditions for exactness. The tutorial then verifies the exactness of a given equation by comparing partial derivatives. It proceeds to find the solution by integrating with respect to x and discusses the inclusion of a function of y. Finally, it compares the derived function with the original equation to finalize the solution, concluding with the solution form.

What is a necessary condition for a differential equation to be considered exact?

• The equation must be linear.

• The equation must be homogeneous.

• The partial of M with respect to x must equal the partial of N with respect to y.

• The partial of M with respect to y must equal the partial of N with respect to x.

In the given differential equation, what is the expression for M(x, y)?

• 2e^(2x) cos(y) - 4y^2 cos(2x)

• 2e^(2x) sin(y) - 4y^2 sin(2x)

• e^(2x) cos(y) + 4y cos(2x)

• e^(2x) sin(y) + 4y sin(2x)

How do you verify that a differential equation is exact?

• By checking if the equation is linear.

• By ensuring the partial derivatives of M and N with respect to x and y are equal.

• By checking if the equation is homogeneous.

• By solving the equation directly.

What is the form of the solution for an exact differential equation?

• f(x, y) = xy

• f(x, y) = c

• f(x, y) = 0

• f(x, y) = x + y

When integrating M with respect to x, what must be included?

• A function of y

• A constant

• A function of x

• A function of both x and y

What substitution is used during the integration process?

• u = 2x

• u = 2y

• u = y

• u = x

What is the result of integrating e^(2x) sin(y) with respect to x?

• e^(2x) sin(y)

• 2e^(2x) sin(y)

• e^(2x) cos(y)

• 2e^(2x) cos(y)

What must h'(y) equal for the solution to be valid?

• A constant

• A function of y

• Zero

• A function of x

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

• e^(2x) sin(y) + 2y^2 cos(2x) = c

• e^(2x) cos(y) + 2y^2 sin(2x) = c

• e^(2x) sin(y) - 2y^2 cos(2x) = c

• e^(2x) cos(y) - 2y^2 sin(2x) = c

Why is there no need to include a constant on the left side of the solution equation?

• Because the constant is zero.

• Because the equation is linear.

• Because the equation is homogeneous.

• Because the constant is already included on the right side.