211_pdfsam_math 54 differential equation solutions odd

211_pdfsam_math 54 differential equation solutions odd -...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Exercises 4.5 (b) For t> 3 π/ 2, g ( t ) 0, and so the given equation becomes homogeneous. Thus, a general solution, y 2 ( t ), is given by (4.1), i.e., y 2 ( t )= y h ( t )=( c 1 cos 2 t + c 2 sin 2 t ) e t . (c) We want to satisfy the conditions y 1 (3 2) = y 2 (3 2) , y 0 1 (3 2) = y 0 2 (3 2) . Evaluating y 1 , y 2 , and their derivatives at t =3 2, we solve the system 2 e 3 2 +2 = c 1 e 3 2 , 0=( c 1 2 c 2 ) e 3 2 c 1 = 2 ( e 3 2 +1 ) , c 2 = ( e 3 2 ) . 43. Recall that the motion of a mass-spring system is governed by the equation my 0 + by 0 + ky = g ( t ) , where m is the mass, b is the damping coefficient, k is the spring constant, and g ( t )isthe external force. Thus, we have an initial value problem y 0 +4 y 0 +3 y =5s in t, y (0) = 1 2 ,y 0 (0) = 0 . The roots of the auxiliary equation, r 2 r +3=0,are r = 3, 1, and a general solution
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.

Ask a homework question - tutors are online