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

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Math 334 Lecture #27 § 6.5: Unit Impulse Functions and Dirac Delta “Functions” Example. How does the solution of the IVP y + 4 y + 4 y = u 1 ( t ) - u 1+ k ( t ) k , y (0) = 0 , y (0) = 0 , behave as k 0 + ? [For k = 1, the IVP is that of the example in the Lecture #32.] [See Maple worksheet for animation of solution as k 0 + .] 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 , ): I ( f k ) = 0 u 1 ( t ) - u 1+ k ( t ) k dt = 1 k 1+ k 1 dt = 1 for k > 0 . What is the limit of the impulse of f k as k 0 + ? lim k 0 + I ( f k ) = 1 . For a fixed value of t 0, what is the limit of f k ( t ) as k 0 + ? It is the “function” δ ( t - 1) = lim k 0 + f k ( t ) = 0 if t = 1 , if t = 1 . What is the impulse of the limit “function” δ ( t - 1)? I ( δ ( t - 1) ) = 0 δ ( t - 1) dt = 0 .
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