HW5 - MA341 Homework 5 4.1 3. The derivatives of y are: y =...

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: MA341 Homework 5 4.1 3. The derivatives of y are: y = 6 cos 3 t- 3 sin 3 t, y 00 =- 18 sin 3 t- 9 cos 3 t. Thus, 2 y 00 + 18 y = 2(- 18 sin 3 t- 9 cos 3 t ) + 18(2 sin 3 t + cos 3 t ) =- 36 sin 3 t- 18 cos 3 t + 36 sin 3 t + 18 cos 3 t = 0 . Substitution into the initial conditions yields: y (0) = 2 sin(0) + cos(0) = 1 , y (0) = 6 cos(0)- 3 sin(0) = 6 . Hence, y = 2 sin 3 t + cos 3 t is a solution to the initial value problem. Moreover, y ( t ) = 2 sin 3 t + cos 3 t = A sin(3 t + ) = A sin 3 t cos + A cos 3 t sin . Then, A cos = 2 and A sin = 1. Consequently, = arctan(1 / 2) and A = 5. We conclude that max- <t< | y ( t ) | = A = 5. 5. The derivatives of y = e- 2 t sin( 2 t ) are: y ( t ) =- 2 e- 2 t sin( 2 t ) + 2 e- 2 t cos( 2 t ) , y 00 ( t ) = 2 e- 2 t sin( 2 t )- 4 2 e- 2 t cos( 2 t ) . Substitution into (3) gives: my 00 + by + ky = y 00 + 4 y + 6 y = 2 e- 2 t sin( 2 t )- 4 2 e- 2 t cos( 2 t ) + 4 {- 2 e- 2 t sin( 2 t ) + 2 e- 2 t cos( 2 t ) } + 6 e- 2 t sin( 2 t ) = 0 . We also have lim t e- 2 t sin( 2 t ) = 0. 7. Let y ( t ) = A cos t + B sin t , where = 5. Then, y ( t ) =- 5 A sin 5 t + 5 B cos 5 t, y 00 ( t ) =- 25 A cos 5 t- 25 B sin 5 t. Next, we substitute these expressions into y 00 + 2 y + 4 y = 3 sin 5 t , collect the terms and match coefficients: 3 sin 5 t = y 00 + 2 y + 4 y = (- 21 B- 10 A ) sin 5 t + (- 21 A + 10 B ) cos 5 t. MA341 Homework 5 Therefore, 3 =- 21 B- 10 A, 0 =- 21 A + 10 B....
View Full Document

This homework help was uploaded on 04/16/2008 for the course MA 341 taught by Professor Schecter during the Spring '08 term at N.C. State.

Page1 / 6

HW5 - MA341 Homework 5 4.1 3. The derivatives of y are: y =...

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