Cauchy Euler equation Problem FEReview10m The Wronskian of the functions y 1 t

Cauchy euler equation problem fereview10m the

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(Cauchy-Euler equation) Problem FEReview.10m The Wronskian of the functions y 1 ( t ) = e 2 t and y 2 ( t ) = 2 t a e 2 t , where a is a real number, is W [ y 1 , y 2 ] ( t ) = 2 e 4 t . The value of the parameter a is (a) - 1 / 2 (b) 1/2 (c) 1 (d) 3/2 Problem FEReview.11m A specific spring-mass system is described by the equation my 00 + γy 0 + ky = 0 , with mass m = 8 kg and spring constant k = 2 kg/s 2 . The value of the damping coefficient γ for which the system changes its behavior from underdamped to overdamped is: (a) 6 kg/s (b) 8 kg/s (c) 10 kg/s (d) 12 kg/s Problem FEReview.12m e - t and e 2 t are fundamental solutions of the homogeneous equation corre- sponding to y 00 - y 0 - 2 y = te 2 t . Which of the following is the correct form of the particular solution ( A and B are two constants)? (a) Y ( t ) = At 2 e 2 t (b) Y ( t ) = Ate - t (c) Y ( t ) = ( At 2 + B ) e 2 t (d) Y ( t ) = ( At + B ) te 2 t 3
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Problem FEReview.13m Which of the following piece-wise continuous functions is drawn on the picture below? (a) f ( t ) = t 2 for 0 t < 1 1 . 5 for 1 < t < 2 1 . 375 for t = 2 3 . 75 - 1 . 25 t for 2 < t 3 (b) f ( t ) = t 2 for 0 t 1 1 . 5 for 1 < t < 2 1 . 375 for t = 2 3 . 75 - 1 . 25 t for 2 < t 3 (c) f ( t ) = t 2 for 0 t 1 1 . 5 for 1 < t < 2 1 . 25 for t = 2 3 . 75 - 1 . 25 t for 2 < t 3 (d) f ( t ) = t 2 for 0 t 1 1 . 5 for 1 < t < 2 1 . 375 for t = 2 3 . 75 - t for 2 < t 3 Problem FEReview.14m The Laplace transform Y ( s ) of the solution of the initial value problem y 00 - y 0 - 2 y = 0 , y (0) = 0 , y 0 (0) = 2 is: (a) Y ( s ) = 2 / ( s 2 - 2 s - 2) (b) Y ( s ) = 2 / ( s 2 - 2 s - 4) (c) Y ( s ) = 2 / ( s - 2) 2 (d) Y ( s ) = 2 / [( s + 1) ( s - 2)] Problem FEReview.15m The inverse Laplace transform of F ( s ) = 5 s 2 + 2 s + 6 is: (a) 5 cos( 5 t ) (b) 5 e t sin( 5 t ) (c) 5 e - t sin( 5 t ) (d) 5 sin( 5 t ) 4
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Free Response Problems Problem FEReview.16 Consider the following differential equation: t 2 dy dt = y - ty (1) (a) Write the equation in the standard form and identify point(s) where the conditions of exis- tence and uniqueness theorem (EUT) are not satisfied. (b) Find the the constant solutions, if any. Next, obtain the general solution of the equation. What is the behavior of the general solution at the point(s) where the conditions of EUT are not satisfied? (c) Find the solutions of the initial value problems: (i) y (1) = 1 and (ii) y ( - 1) = - 1 and infer their respective ranges of validity. (d) Find the maxima/minima of the solutions (i) and (ii) from subproblem (c), if any. Find the limit of the solution (i) for t → ∞ and of the solution (ii) for t → -∞ and sketch the graphs of the solutions in both cases (i) and (ii).
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