exact Equations

Question 1

What is an exact equation?

Answer 1

An exact equation is a first-order differential equation of the form M(x, y)dx + N(x, y)dy = 0 where ∂M/∂y = ∂N/∂x.

Question 2

True or False: An exact equation can always be solved by direct integration.

Answer 2

True.

Question 3

Fill in the blank: An exact equation must satisfy the condition _____ for it to be considered exact.

Answer 3

∂M/∂y = ∂N/∂x.

Question 4

What is the general solution of an exact equation?

Answer 4

The general solution is given by the level curves of a function F(x, y) such that dF = Mdx + Ndy.

Question 5

What does it mean if an equation is not exact?

Answer 5

It means that the condition ∂M/∂y ≠ ∂N/∂x is not satisfied.

Question 6

How can you make a non-exact equation exact?

Answer 6

You can multiply the equation by an integrating factor.

Question 7

What is an integrating factor?

Answer 7

An integrating factor is a function μ(x, y) that, when multiplied with a non-exact equation, makes it exact.

Question 8

True or False: The integrating factor depends only on x for all non-exact equations.

Answer 8

False.

Question 9

What is the first step in solving an exact equation?

Answer 9

Verify that the equation is exact by checking the condition ∂M/∂y = ∂N/∂x.

Question 10

Multiple Choice: Which of the following is a method to solve an exact equation? A) Direct integration B) Substitution C) Factoring D) Integration by parts

Answer 10

A) Direct integration.

Question 11

What is the role of the function F(x, y) in an exact equation?

Answer 11

F(x, y) represents the potential function whose differential gives the exact equation.

Question 12

True or False: If M and N are functions of both x and y, the equation may still be exact.

Answer 12

True.

Question 13

What is the significance of the total differential in exact equations?

Answer 13

The total differential dF = Mdx + Ndy must equal zero for the solution curves.

Question 14

Fill in the blank: If F(x, y) is a solution to an exact equation, then the level curves of F correspond to _____ of the original equation.

Answer 14

solutions.

Question 15

What is the relationship between the potential function F and the exact equation?

Answer 15

The potential function F satisfies ∂F/∂x = M and ∂F/∂y = N.

Question 16

Multiple Choice: Which of the following is not a characteristic of an exact equation? A) Linear B) Nonlinear C) Can be solved using integrating factors D) Has a unique solution

Answer 16

D) Has a unique solution.

Question 17

What does it mean for an exact equation to have a unique solution?

Answer 17

It means that there is only one curve in the solution space that satisfies the equation for given initial conditions.

Question 18

True or False: An exact equation can have infinitely many solutions.

Answer 18

True.

Question 19

What is the condition for the existence of an integrating factor?

Answer 19

The condition varies; it may depend on the functions M and N and may not always exist.

Question 20

Fill in the blank: The existence of an integrating factor can often be tested using _____ conditions.

Answer 20

exactness.

Question 21

What is the main advantage of using exact equations in differential equations?

Answer 21

They can be solved directly without the need for additional techniques.

Question 22

Multiple Choice: Which of the following techniques is commonly used to find integrating factors? A) Variation of parameters B) Laplace transforms C) Bernoulli’s equation D) Multiplication by a function

Answer 22

D) Multiplication by a function.

Question 23

What is the significance of the equality ∂M/∂y = ∂N/∂x?

Answer 23

It ensures that the differential form is exact and can be integrated to find a solution.

Question 24

True or False: All first-order differential equations are exact.

Answer 24

False.