exact Equations Flashcards
What is an exact equation?
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.
True or False: An exact equation can always be solved by direct integration.
True.
Fill in the blank: An exact equation must satisfy the condition _____ for it to be considered exact.
∂M/∂y = ∂N/∂x.
What is the general solution of an exact equation?
The general solution is given by the level curves of a function F(x, y) such that dF = Mdx + Ndy.
What does it mean if an equation is not exact?
It means that the condition ∂M/∂y ≠ ∂N/∂x is not satisfied.
How can you make a non-exact equation exact?
You can multiply the equation by an integrating factor.
What is an integrating factor?
An integrating factor is a function μ(x, y) that, when multiplied with a non-exact equation, makes it exact.
True or False: The integrating factor depends only on x for all non-exact equations.
False.
What is the first step in solving an exact equation?
Verify that the equation is exact by checking the condition ∂M/∂y = ∂N/∂x.
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
A) Direct integration.
What is the role of the function F(x, y) in an exact equation?
F(x, y) represents the potential function whose differential gives the exact equation.
True or False: If M and N are functions of both x and y, the equation may still be exact.
True.
What is the significance of the total differential in exact equations?
The total differential dF = Mdx + Ndy must equal zero for the solution curves.
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.
solutions.
What is the relationship between the potential function F and the exact equation?
The potential function F satisfies ∂F/∂x = M and ∂F/∂y = N.
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
D) Has a unique solution.
What does it mean for an exact equation to have a unique solution?
It means that there is only one curve in the solution space that satisfies the equation for given initial conditions.
True or False: An exact equation can have infinitely many solutions.
True.
What is the condition for the existence of an integrating factor?
The condition varies; it may depend on the functions M and N and may not always exist.
Fill in the blank: The existence of an integrating factor can often be tested using _____ conditions.
exactness.
What is the main advantage of using exact equations in differential equations?
They can be solved directly without the need for additional techniques.
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
D) Multiplication by a function.
What is the significance of the equality ∂M/∂y = ∂N/∂x?
It ensures that the differential form is exact and can be integrated to find a solution.
True or False: All first-order differential equations are exact.
False.
