nagle_differential_equations_ISM_Part34

nagle_differential_equations_ISM_Part34 - Exercises 4.9 It...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Exercises 4.9 It follows that period = 2 π ω = 2 π 5 natural frequency = ω 2 π = 5 2 π . A general solution, given in (4) in the text, becomes y ( t ) = C 1 cos ωt + C 2 sin ωt = C 1 cos 5 t + C 2 sin 5 t. We find C 1 and C 2 from the initial conditions. y (0) = ( C 1 cos 5 t + C 2 sin 5 t ) t =0 = C 1 =- 1 / 4 y (0) = (- 5 C 1 sin 5 t + 5 C 2 cos 5 t ) t =0 = 5 C 2 =- 1 ⇒ C 1 =- 1 / 4 C 2 =- 1 / 5 . Thus, the solution to the initial value problem is y ( t ) =- 1 4 cos 5 t- 1 5 sin 5 t. The amplitude of the motion therefore is A = q C 2 1 + C 2 2 = r 1 16 + 1 25 = √ 41 20 . Setting y = 0 in the above solution, we find values of t when the mass passes through the point of equilibrium.- 1 4 cos 5 t- 1 5 sin 5 t = 0 ⇒ tan 5 t =- 5 4 ⇒ t = πk- arctan(5 / 4) 5 , k = 1 , 2 ,.... (Time t is nonnegative.) The first moment when this happens, i.e., the smallest value of t , corresponds to k = 1. So, t = π- arctan(5 / 4) 5 ≈ . 45 (sec) . 4. The characteristic equation in this problem, r 2 + br + 64 = 0, has the roots r =- b ± √ b 2- 256 2 . Substituting given particular values of b , we find the roots of the characteristic equation and solutions to the initial value problems in each case. 161 Chapter 4 b = 0 b = 0 b = 0. r = ± √- 256 2 = ± 8 i. A general solution has the form y = C 1 cos 8 t + C 2 sin 8 t . Constants C 1 and C 2 can be found from the initial conditions....
View Full Document

This note was uploaded on 02/12/2011 for the course MA 221 taught by Professor Mazmani during the Spring '08 term at Stevens.

Page1 / 5

nagle_differential_equations_ISM_Part34 - Exercises 4.9 It...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online