c11mudd

# c11mudd - 18.03 Muddy Card responses March 3 2006 1 I...

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

18.03 Muddy Card responses, March 3, 2006 1. I confused a number of people by dividing through m ¨ x + bx ˙ + kx = F ext ( t ) and ¨ miraculously getting x + bx ˙ + kx = q ( t ). What I meant to say was that by dividing through you make the coeﬃcient of x equal to 1 and replace the others by b/m and ¨ k/m ; but then I’ll change the meaning of the symbol b so that it means the damping constant divided by the mass, and of k so that it means the spring constant divided by the mass. 2. Here’s the proof that if x 1 and x 2 are solutions of the homogeneous, linear second ¨ order ODE x + b ( t ) ˙ x + k ( t ) x = 0 then so is x = c 1 x 1 + c 2 x 2 . Plug this x into the equation: k ( t ) x = k ( t ) c 1 x 1 + k ( t ) c 2 x 2 b ( t ) ˙ x = b ( t ) c 1 x ˙ 1 + k ( t ) c 2 x ˙ 2 x ¨ ¨ ¨ = c 1 x 1 + c 2 x 2 ¨ x 1 + b ( t ) ˙ x 1 + k ( t ) x 1 ) + c 2 x + b ( t ) ˙ x + k ( t ) x = c 1 x 2 + b ( t ) ˙ x 2 + b ( t ) x 2 ) and both terms on the right are zero, by assumption. As the notation indicates, this
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 01/31/2011 for the course MAT 17A taught by Professor Staff during the Winter '08 term at UC Davis.

Ask a homework question - tutors are online