Differential Equations: Definitions and Terminology (Level 1 of 4)

This video tutorial introduces differential equations, highlighting their significance in various fields. It covers essential calculus prerequisites, discusses functions and derivatives, and explains how to solve basic differential equations. The tutorial classifies differential equations by type (ordinary or partial), order, and linearity, providing examples and emphasizing the importance of understanding these classifications for solving differential equations effectively.

Which of the following fields does NOT prominently use differential equations?

• Biology

• Literature

• Physics

• Engineering

What is the derivative of the function y = e^(x^2)?

• x^2e^(x)

• e^(2x)

• 2xe^(x^2)

• 2x^2e^(x^2)

What is an ordinary differential equation (ODE)?

• An equation with partial derivatives

• An equation with ordinary derivatives with respect to a single variable

• An equation with multiple independent variables

• An equation with no derivatives

Which notation is typically used for the first three derivatives in ODEs?

• Dot notation

• Prime notation

• Leibniz notation

• Subscript notation

What determines the order of a differential equation?

• The number of terms

• The highest derivative present

• The number of variables

• The highest power of the variable

What is the normal form of a first order ODE?

• y' = f(x, y)

• y'' = f(x, y')

• y = f(x)

• y' = x + y

Which of the following is a characteristic of a linear ODE?

• It contains trigonometric functions of the dependent variable

• The dependent variable and its derivatives are of the first degree

• It has no constant terms

• The dependent variable is squared

Which of the following is NOT a linear ODE?

• y'' + y = sin(x)

• y' + xy = 0

• y' + y^2 = 0

• y'' + 3y' + 2y = 0

What makes an ODE nonlinear?

• It is of the first order

• It has no derivatives

• It contains nonlinear functions of the dependent variable or its derivatives

• It has more than one independent variable

Which of the following is an example of a nonlinear term in an ODE?

• e^y

• y''

• y'

• x