In the damped harmonic oscillator, we assume that the coefficients m, b, and k are positive. However, the rationale underlying the guess-and-test method made no such assumption, and the same analytic technique can be used if some or all of the coefficients of the equation are negative. In Exercises 5 and 6, make the same graphs and perform the same calculations as were specified in Exercises 1—4. What is different in this case?
6.
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Chapter 2 Solutions
Differential Equations
- In Exercises 49–54, solve for y in terms of t or x, as appropriate.arrow_forwardExercises 38–40 will help you prepare for the material covered in the first section of the next chapter. In Exercises 38-39, simplify each algebraic expression. 38. (-9x³ + 7x? - 5x + 3) + (13x + 2r? – &x – 6) 39. (7x3 – 8x? + 9x – 6) – (2x – 6x? – 3x + 9) 40. The figures show the graphs of two functions. y y 201 10- .... -20- flx) = x³ glx) = -0.3x + 4x + 2arrow_forwardFor Exercises 41–54, write the equation in the form (x – h + (y – k)' = c. Then if the equation represents a circle, identify the center and radius. If the equation represents a degenerate case, give the solution set. (See Examples 3-4) 41. x + y? - + 6x – 2y + 6 = 0 42. x + y + 12x – 14y + 84 = 0 43. x + y 22x + 6y + 129 = 0 44. x + y? - 10x + 4y – 20 = 0 45. x + y - 20y – 4 = 0 46. x + y + 22x – 4 = 0 47. 10x + 10y² – 80x + 200y + 920 = 0 (Hint: Divide by 10 to make the x and y2 term coefficients equal to 1.) 48. 2x + 2y? 32x + 12y + 90 = 0 49. x + y? – 4x – 18y + 89 = 0 50. x + y - 10x – 22y + 155 = 0 51. 4x? + 4y 20y + 25 = 0 52. 4x + 4y? 12x + 9 = 0 53. x + y - x - 3 3 -y - 54. x + v? 2 = 0 = 0 9. -X - 2 4 3 3arrow_forward
- *There are two solutions to the equation (x – 1)3 – 9 = 0. Find both and enter them below as whole numbers.arrow_forwardYou may find it helpful to review the information in the Reasonable Answers box from this section before answering Exercises 13–16. Verify that the solutions you found to Exercise 9 are indeed homogeneous solutions.arrow_forwardQuestion 2 Evaluate 2x /xb +c + D. 9.arrow_forward
- Exercises 7–12: Determine whether the equation is linear or nonlinear by trying to write it in the form ax + b = 0. 7. 3x – 1.5 = 7arrow_forwardFor Exercises 54–60, a. List all possible rational roots or rational zeros. b. Use Descartes's Rule of Signs to determine the possible number of positive and negative real roots or real zeros. c. Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. d. Use the quotient from part (c) to find all the remaining roots or zeros. 54. f(x) = x' + 3x² – 4 55. flx) = 6x + x² – 4x + 1 %3D 56. 8x - 36xr? + 46x - 15 = 0 57. 2x + 9x2 - 7x + 1 = 0 58. x* - x - 7x2 + x + 6 = 0 59. 4x* + 7x - 2 = 0 60. f(x) = 2x* + x³ – 9x² – 4x + 4arrow_forwardIn Exercises 9–12, find all real solutions to the equations.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage