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chapter6.page07 - f ( t ), we need to change f ( t ) into...

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3. THE PIECEWISE DEFINED FUNCTIONS 7 Step Three: Solve the algebraic equation and apply the in- verse Laplace transform, using Mathcad we have b x ( s ) = A - 1 b ( s ) = 1 s 2 - 8 s + 11 " (s - 5)(s 3 +6) s 3 - 4 2 - s s - 1 ( s - 3)(2 - s ) s - 1 - s 3 +6 s 3 # x ( t ) = x(t) y ( t ) = - 15 11 t 2 - 174 121 t - 1062 1331 + e 4t ± 3742 1331 cosh( 5t) - 2334 6655 sinh( 5t) - e t - 3 11 t 2 - 48 121 t - 318 1331 - e 4 t ± 695 2662 cosh( 5 t ) + 8143 1331 sinh( 5 t ) - 1 2 e t a 3. The Piecewise defined Functions When solving an ODE a n x ( n ) + a n - 1 x ( n - 1) + ··· + a 1 x 0 + a 0 x = f ( t ) with multiple defined function
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Unformatted text preview: f ( t ), we need to change f ( t ) into linear combinations of step functions u a ( t ). Here u a ( t ) = u ( t-a ) = ‰ 0 if t < a 1 if t ≥ a and u ( t ) = ‰ 0 if t < 1 if t ≥ is called Heaviside function or unit step function . We have graphed some step functions, Figure 1. Step Functions...
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This note was uploaded on 10/04/2011 for the course MAP 4231 taught by Professor Dr.han during the Spring '11 term at UNF.

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