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practice_exam3

# practice_exam3 - x = 0 where 4 represents the spring...

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Math 2700, Fall 2011, Practice Sheet for Exam 3 Find general analytic solutions to the following differential equations. . 1. y 00 - 3 y 0 + 2 y = 0 2. y 00 + y 0 + y = 0 3. y 00 + 2 y 0 - 15 y = 0 4. y 00 + 4 y = 0 5. y 00 - 4 y = 0 6. d~v dt = A~v , where A = 0 1 - 2 3 Solve the following initial value problems (use your work above!) 7. y 00 - 4 y = 0 , y (0) = 1 , y 0 (0) = 3 8. y 00 + y 0 + y = 0 , y (0) = 4 , y 0 (0) = 0 9. d~v dt = A~v, ~v = 2 1 , where A = 0 1 - 2 3 Determine whether the fixed point at the origin is a sink, souce, node, spiral sink, or spiral source for the following systems: 10. d~v dt = A~v , where A = 3 1 2 3 11. d~v dt = A~v , where A = 0 - 1 2 - 1 Modeling/Applications with the harmonic oscillator: 12. Suppose we have a mass on a spring described by the differential equation x 00 + dx 0 + 4 x = 0
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Unformatted text preview: x = 0 where 4 represents the spring constant and d the damping coeﬃcient (a positive number). For which values of d is the system undamped, underdamped, or overdamped? 13. Suppose we have a mass on a spring described by the diﬀerential equation x 00 + 4 x = 0 (1) Find the natural resonant frequency of this oscillator. Call this ω . (2) Find any speciﬁc solution to the forced oscillator equation x 00 + 4 x = cos( ωt ) (3) Solve for, using the same ω , any speciﬁc solution to the forced and damped equation x 00 + 2 x + 4 x = cos( ωt ) 1...
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