Modelling With First Order Equations Flashcards
What is a first order differential equation?
A first order differential equation is an equation that involves the first derivative of a function and the function itself.
True or False: A first order equation can be written in the form dy/dx = f(x, y).
True
Fill in the blank: The general solution of a first order linear differential equation is of the form y = ______.
Ce^(kx) + particular solution
What does the ‘C’ represent in the general solution of a first order differential equation?
The constant of integration.
Which of the following is a method for solving first order differential equations? (A) Separation of variables (B) Quadratic formula (C) Completing the square
A
True or False: The initial value problem for a first order equation provides a specific solution.
True
What is the standard form of a first order linear equation?
dy/dx + P(x)y = Q(x)
Fill in the blank: The integrating factor for the equation dy/dx + P(x)y = Q(x) is given by e^(_________).
∫P(x)dx
What type of solution does the method of separation of variables yield?
A general solution involving a constant.
True or False: All first order differential equations can be solved using the same method.
False
What is the purpose of an integrating factor?
To transform a non-exact equation into an exact one.
Fill in the blank: In the context of first order equations, a ‘particular solution’ is a solution that satisfies both the differential equation and the ______.
initial conditions
Which of the following represents a separable differential equation? (A) dy/dx = y^2 + x (B) dy/dx = xy (C) dy/dx = e^x
B
What is the key feature of a homogeneous first order differential equation?
It can be expressed in the form dy/dx = f(y/x).
True or False: The solution to a first order differential equation can be a function that is not continuous.
False
What does the term ‘exact equation’ refer to in first order differential equations?
An equation that can be expressed as M(x,y)dx + N(x,y)dy = 0, where ∂M/∂y = ∂N/∂x.
Fill in the blank: A first order linear differential equation can be solved using the method of ______.
undetermined coefficients
Which equation represents a first order separable equation?
dy/dx = g(x)h(y)
True or False: The initial condition can affect the uniqueness of the solution for a first order differential equation.
True
What is the typical first step in solving a separable differential equation?
Rearranging the equation to isolate variables.
Fill in the blank: The solution to dy/dx = ky, where k is a constant, is of the form y = ______.
Ce^(kt)
What is the significance of the slope in a first order differential equation?
It represents the rate of change of the dependent variable with respect to the independent variable.
Which of the following is NOT a characteristic of first order differential equations? (A) They involve only first derivatives (B) They can model growth and decay (C) They can only be solved graphically
C
What is the method of characteristics used for?
To solve certain types of first order partial differential equations.
True or False: A first order differential equation can be linear or nonlinear.
True
Fill in the blank: The solution to a first order linear differential equation can often be expressed as a ______.
combination of complementary and particular solutions
