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set18 - $ Set 18 Laplace Transform and IVPs Part 3 Kyle A...

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a39 a38 a36 a37 Set 18: Laplace Transform and IVPs Part 3 Kyle A. Gallivan Department of Mathematics Florida State University Ordinary Differential Equations Fall 2009 1
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a39 a38 a36 a37 General Form ay ′′ + by + cy = f ( t ) , y (0) = y 0 , y (0) = y 0 arrowdblbothv Y ( s ) = ( as + b ) y 0 + ay 0 as 2 + bs + c + F ( s ) as 2 + bs + c where F ( s ) = L{ f } . 2
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a39 a38 a36 a37 Example Ramp loading: f ( t ) continuous f ( t ) discontinuous at two points t = 5 and t = 10 y ′′ + 4 y = f ( t ) y (0) = 0 y (0) = 0 We could solve it using our standard techniques in a piecewise manner. 3
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a39 a38 a36 a37 Ramp Loading f ( t ) = 0 0 t < 5 ( t 5) / 5 5 t < 10 1 t 10 . . . . . . 4
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a39 a38 a36 a37 Ramp Loading 0 5 10 15 20 25 30 -1 -0.5 0 0.5 1 1.5 2 5
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a39 a38 a36 a37 Example y ′′ + 4 y = f ( t ) , y (0) = 0 , y (0) = 0 a = 1 , b = 0 , c = 4 Y ( s ) = 0 s 2 + 4 + F ( s ) s 2 + 4 f ( t ) = 1 5 ( u 5 ( t )( t 5) u 10 ( t )( t 10) ) F ( s ) = 1 5 bracketleftbig e 5 s s 2 e 10 s s 2 bracketrightbig 6
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a39 a38 a36 a37 Example Y ( s ) = 1 5 bracketleftbig 1 s 2 + 4 bracketrightbigbracketleftbig e 5 s e 10 s s 2 bracketrightbig = e
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