quiz5-sol

# quiz5-sol - Name MA 366 Spring 2010 Quiz 5(1(8 points...

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Unformatted text preview: Name: MA 366, Spring 2010, Quiz 5 (1) (8 points) Consider the system 2 u ′′ + γu ′ + u = 0 , where γ is a nonnegative parameter. For which values of γ is the system (a) underdamped, (b) overdamped, and (c) critically damped? Solution: The characteristic equation is 0 = 2 r 2 + γr + 1, which has roots r 1 , r 2 =- γ ± radicalbig γ 2- 4 × 2 × 1 2 × 2 . The system is underdamped if it has two complex roots (i.e., γ 2- 8 < 0 or 0 ≤ γ < 2 √ 2, since γ ≥ 0), it is critically damped if it has two equal roots (i.e., if γ 2- 8 = 0 or γ = 2 √ 2), and it is overdamped if it has two negative real roots (i.e., if γ 2- 8 > 0 or γ > 2 √ 2). (2) (10 points) Determine the general solution of the equation y ′′ + 2 y ′ + y = e t + cos t. Solution: The characteristic equation is 0 = r 2 + 2 r + 1 = ( r + 1) 2 , so the homogeneous solution is y h = c 1 e − t + c 2 te − t ....
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quiz5-sol - Name MA 366 Spring 2010 Quiz 5(1(8 points...

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