Understanding Exact Differential Equations

The video tutorial explains how to solve an exact first-order differential equation. It begins by defining exact differential equations and verifying the equality of partial derivatives. The tutorial then guides through solving the equation by integrating to find the function f(x, y) and determining the function h(y) to reach the final solution.

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

• The integral of M with respect to x equals the integral of N with respect to y.

• The partial derivative of M with respect to y equals the partial derivative of N with respect to x.

• The partial derivative of M with respect to x equals the partial derivative of N with respect to y.

• The sum of M and N equals zero.

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

• Negative two x squared y minus four x squared minus five

• Negative two xy squared minus eight xy

• Negative four x y minus eight x

• Negative x squared y squared minus four x squared y

How do you verify the exactness of a differential equation?

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

• By substituting values into the equation.

• By integrating M and N with respect to x and y respectively.

• By solving for f(x, y) directly.

What is the result of differentiating M with respect to y?

• Negative four x y minus eight x

• Negative two x squared y minus four x squared

• Negative x squared y squared minus four x squared y

• Negative five y

What is the first step in solving an exact differential equation?

• Differentiate N with respect to x.

• Differentiate M with respect to y.

• Integrate M with respect to x.

• Integrate N with respect to y.

Why do we add a function of y when integrating M with respect to x?

• To account for the x terms in f(x, y).

• To simplify the integration process.

• To ensure the solution is exact.

• To account for the y terms in f(x, y).

What does h'(y) equal in the solution process?

• Negative two x squared y

• Negative five

• Negative four x squared

• Zero

How do you find h(y) from h'(y)?

• By integrating h'(y) with respect to y.

• By setting h'(y) equal to zero.

• By substituting h'(y) into the original equation.

• By differentiating h'(y) with respect to y.

What is the final form of f(x, y) in the solution?

• Negative two x squared y minus four x squared minus five

• Negative x squared y squared minus four x squared y minus five y

• Negative two xy squared minus eight xy

• Negative four x y minus eight x

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

• f(x, y) = xy

• f(x, y) = x + y

• f(x, y) = c

• f(x, y) = 0