M334Lec27

# M334Lec27 - Math 334 Lecture#27 Â 6.5 Unit Impulse...

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Unformatted text preview: Math 334 Lecture #27 Â§ 6.5: Unit Impulse Functions and Dirac Delta âFunctionsâ Example. How does the solution of the IVP y 00 + 4 y + 4 y = u 1 ( t )- u 1+ k ( t ) k , y (0) = 0 , y (0) = 0 , behave as k â + ? [For k = 1, the IVP is that of the example in the Lecture #26.] [See Maple worksheet for animation of solution as k â + .] An Idealized Impulse Force. The discontinuous function f k ( t ) = u 1 ( t )- u 1+ k ( t ) k describes the constant force of 1 /k applied on the interval [1 , 1 + k ]. [Sketch the graph of the discontinuous forcing function for several values of k , especial small values.] The impulse (or strength) of f k is its integral over [0 , â ): I ( f k ) = Z â u 1 ( t )- u 1+ k ( t ) k dt = 1 k Z 1+ k 1 dt = 1 for k > . What is the limit of the impulse of f k as k â + ? lim k â + I ( f k ) = 1 . For a fixed value of t âĽ 0, what is the limit of f k ( t ) as k â + ? It is the âfunctionâ Î´ ( t- 1) = lim k â + f k ( t ) = if t 6 = 1 , â if t = 1 ....
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M334Lec27 - Math 334 Lecture#27 Â 6.5 Unit Impulse...

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