{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

c18 - 18.03 Class 18 Review of constant coefficient linear...

Info iconThis preview shows pages 1–2. 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 Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 18.03 Class 18, March 20, 2006 Review of constant coefficient linear equations: Big example, superposition, and Frequency Response [1] Example. x" + 4x = 0 PLEASE KNOW the solution to the homogeneous harmonic oscillator x" + omega^2 x = 0 are sinusoids of circular frequency omega ! Here, a cos(2t) + b sin(2t). In the real example I drive it: x" + 4x = t cos(2t) . The complex equation is z" + 4z = t e^{2it} . If it weren't for the t we could try to apply ERF: p(s) = s^2 + 4, p(2i) = -4 + 4 = 0 , though, so it doesn't apply; we do have the resonance response formula, which gives z_p = t e^{2it}/p'(2i) = -(it/4) e^{2it} so x_p = (t/4) sin(2t) . But there is a t there. We should then use "Variation of Parameters": Look for solutions of the form z = e^{2it} u for u an Unknown function. 4] z = e^{2it} u 0] z' = e^{2it} ( u' + 2i u ) 1] z" = e^{2it} ( u" + 2i u' + 2i u' + (2i)^2 u )------------------------------------------------------ e^{2it} t = e^{2it} ( u" + 4i u' + (4-4) u ) so u" + 4i u' = t Reduction of order: v = u' , v' + 4i v = t ; Use undetermined coefficients: v = at + b 4i] v = at + b v' = a----------------...
View Full Document

{[ snackBarMessage ]}

Page1 / 3

c18 - 18.03 Class 18 Review of constant coefficient linear...

This preview shows document pages 1 - 2. Sign up to view the full document.

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