lt2 - Boise State University Department of Electrical and...

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Department of Electrical and Computer Engineering ECE225 – Circuit Analysis and Design The Laplace Transform II Reading Assignment : Read Sec. 15.4 Lecture Objectives : 1. To derive the Laplace transforms of the derivative and integral of a time function. 2. To apply the partial-fraction technique in the solution of linear differential equations. Property 5: Derivative of a Time Function L{ f ( t ) } = Z 0 f ( t ) e - st dt = F ( s ) L ' f 0 ( t ) = L df dt ± = sF ( s ) - f (0 - ) Proof : L ' f 0 ( t ) = L df dt ± = Z 0 - df dt e - st dt = Z 0 - e - st df = [ f ( t ) e - st ] 0 - - Z 0 - f ( t ) d ( e - st ) = [ f ( ) e - s - f (0 - ) e - s 0 - ] + s Z 0 - f ( t ) e - st dt = sF ( s ) - f (0 - ) Higher Derivatives of a Time Function : L ' f 00 ( t ) = L d dt £ f 0 ( t ) / ± = s L ' f 0 ( t ) - f 0 (0 - ) = s [ sF ( s ) - f (0 - )] - f 0 (0 - ) = s 2 F ( s ) - sf (0 - ) - f 0 (0 - ) L ' f 000 ( t ) = L d dt £ f 00 ( t ) / ± = s L ' f 00 ( t ) - f 00 (0 - ) = s [ s 2 F ( s ) - sf (0 - ) - f 0 (0 - )] - f 00 (0 - ) = s 3 F
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lt2 - Boise State University Department of Electrical and...

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