Chapter 4: Differential Equations and Transforms

Question 1

A differential equation is a mathematical expression combining a function (e.g., y = f (x)) and one or more of its derivatives. The order of a differential equation is ____.

Answer 1

the highest derivative in it.

Question 2

A linear differential equation can be written as a sum of multiples of the function and its derivatives. If the multipliers are scalars, the differential equation is said to have ____

Answer 2

constant coefficients.

Question 3

If the forcing function is ____, the differential equation is said to be homogeneous.

Answer 3

zero

Question 4

If the function or one of its derivatives is raised to some power (other than one) or is embedded in another function (e.g., y embedded in sin y or ey), the equation is said to be ____.

Answer 4

nonlinear

Question 5

For linear, homogenous differential equations, the characteristic equation is:

Answer 5

simply the polynomial formed by replacing all derivatives with variables raised to the power of their respective derivatives.

Question 6

Underdamped

Answer 6

• damping ratio less than 1
• 2 complex roots
• a^2 < 4b (form of (α + iβ) and (α − iβ))

Question 7

Overdamped

Answer 7

• damping ratio greater than 1
• 2 distinct real different roots (zeros)
• a^2 > 4b

Question 8

Critically damped

Answer 8

• damping ratio = 1
  -2 identical real roots (zeros)
• a^2 = 4b

Question 9

In a nonhomogeneous differential equation, the sum of derivative terms is equal to a ____ forcing function of the independent variable

Answer 9

nonzero.