18.03 Lecture Notes - Lecture 16: Frequency Response, Linear Model, Cyan
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[5: supplement] other systems [not done in lecture] [1] i promised on monday to show you what you can do if a is not constant in p(d)x = a e^{rt} . Example: 3x" + 8x" + 6x = (t^2 + 1) e^{-t} . Try for a solution of the form x_p = u e^{-t} for some u . This is what led us to the erf; but now u is allowed to be nonconstant. 8 ] x_p" = (u" - u) e^{-t} 3 ] x_p" = (u" - u" - u" + u) e^{-t} (t^2 + 1) e^{-t} = (3u" + 2u" + u) e^{-t} Cancel the e^{-t} : 3u" + 2u" + u = t^2 + 1. In fact, by an incredible stroke of luck, we have already solved it! u_p = t^2 - 4t + 3 so x_p = u_p e^{-t} = ( t^2 - 4t + 3 ) e^{-t}
