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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 . ...
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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.
- Spring '06