Section 3.5

# Section 3.5 - Notes for Day : . : Complex Roots Last time,...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Notes for Day : . : Complex Roots Last time, we made the observation that if the characteristic equation ak + bk + c = of a second order linear homogeneous di erential equation had complex roots, then: the roots had to be conjugates of one another: k = + i, and k = - i. the real part of the roots matched up with the function e t , while the imaginary part of the roots matched up with the functions sin(t) and cos(t). ( is is because the i changes the signs of the Maclaurin expansion when squared.) As promised, we'll work with some examples before moving on: Exercise ab Find the general solution to the di erential equation y + y + y = . Exercise ab Find the general solution to the di erential equation y + y = . Example (almost) ( is is not exactly the problem that's worked out in the book.) Solve the initial value problem y + y + y = y( ) = y ( ) = Amplitude and Phase Shi Using trig identities, it is possible to express the general solution to these equations with only a single cosine function. is latter form is especially useful in the eld of signal processing. at is, we can rewrite: y(t) = c e t sin(t) + c e t cos(t) in the form using the formulas R= c +c c tan = , c y(t) = Re t cos(t + ) which are derived on page . Warning: ere is a subtle di culty in the second equation: Suppose that we have found c = t, so that tan = t. ere c are two possible quadrants for , and only one is correct! In the next example, we discuss how to resolve this di culty. Example Convert the solution y(t) = e -t (cos t + sin t) into the form y(t) = Re t cos(t + ). Example Solve the initial value problem y + y = , y( ) = - , y ( ) = - , and convert the solution into the form y(t) = Re t cos(t + ). e term Re t is referred to as the amplitude of the resulting periodic function, and the constant is referred to as the phase shi . ...
View Full Document

## This note was uploaded on 04/05/2008 for the course MATH 2214 taught by Professor Edesturler during the Spring '06 term at Virginia Tech.

### Page1 / 2

Section 3.5 - Notes for Day : . : Complex Roots Last time,...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online