Linear First Order Differential Equations Flashcards
What is the general form of a first-order linear differential equation?
The general form is dy/dx + P(x)y = Q(x).
True or False: The term P(x) in a first-order linear differential equation can be a function of y.
False.
Fill in the blank: The solution to a first-order linear differential equation can be found using an _______ factor.
integrating
What is the integrating factor for the equation dy/dx + 2y = 3?
e^(2x)
Multiple Choice: Which of the following is a first-order linear differential equation? A) y’’ + 3y’ + 2y = 0 B) dy/dx + y = sin(x) C) y’’’ + 4y = x
B) dy/dx + y = sin(x)
What is the standard form of the equation dy/dx = 4y + 2?
dy/dx - 4y = 2.
True or False: The solution to a first-order linear differential equation is unique given an initial condition.
True.
Fill in the blank: The first step in solving a first-order linear differential equation is to find the _______.
integrating factor
What is the solution to the equation dy/dx + 3y = 6?
y(x) = 2 - Ce^(-3x).
Multiple Choice: In the equation dy/dx + xy = x, what is P(x)? A) x B) 1 C) 0
A) x
What is the integrating factor for the equation dy/dx - 5y = 10?
e^(-5x)
True or False: The term Q(x) represents the non-homogeneous part of the equation.
True.
Which equation represents a first-order linear differential equation?
dy/dx + 7y = 4x.
Fill in the blank: The solution to a first-order linear differential equation can often be expressed in terms of _______.
exponentials
Multiple Choice: What is the solution to dy/dx + y = 0? A) y = Ce^x B) y = Ce^(-x) C) y = Cx
B) y = Ce^(-x)
What does the term ‘linear’ refer to in first-order linear differential equations?
It refers to the fact that the dependent variable and its derivatives appear to the first power.
True or False: A first-order linear differential equation can have variable coefficients.
True.
What is the standard form of the equation dy/dx + y/x = 3?
dy/dx + (1/x)y = 3.
Fill in the blank: The solution to dy/dx + 4y = 8 is y(x) = _______.
2 - Ce^(-4x)
Multiple Choice: In the equation dy/dx = 2y + sin(x), which term is the dependent variable? A) dy B) y C) x
B) y
What is the solution for the initial value problem dy/dx + y = 1 with y(0) = 0?
y(x) = 1 - e^(-x).
True or False: The general solution of a first-order linear differential equation contains a constant of integration.
True.
What is the form of the solution for the equation dy/dx + 3y = 0?
y(x) = Ce^(-3x).
Fill in the blank: The equation dy/dx + 1 = 2y can be rewritten as _______.
dy/dx - 2y = -1.
