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【正文】 ?? 83r ??? 83i ???f(t) has a form te)]t2s in (83)t2c os (83[21 ??)]t2s i n ()t2[ c o s (e16 3e16 1e81))5s)(1s)(5s2s( s3(163。 21 ????Solve Q(s) has unrepeated linear factor (s5) and (s+1), and an unrepeated quadratic factor (s22s+5), which is (s1)2+4. For (s5)? f(t) has a form t5t539。?Chapter 5 Laplace Transforms . Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Inverse Laplace Transforms Case 2 : If k ? 2 and Q(s) contains the linear factor (sa)k, but not (sa)k+1 , then the corresponding term in f(t) is where )s(Q)as)(s(P)s(H k??at1k2k39。. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Inverse Laplace Transforms Consider the problem of finding , where P(s) and Q(s) are polynomials Having no mon factor and Q(s) has higher degree than P(s). ))s(Q )s(P(163。163。 ?? yy)()()()()()]0()0()[( 39。2239。Subsidiary equation (輔助方程式 ) )()0()0()()( 39。2 sRbYysYaysyYs ??????)(),( rRyY ??163。39。039。 ts ??? s in1)1(221 ???163。163。 163。 163。? ?? ???? ?/0s/2 s ine1 1[ f ( t ) ] t dte st163。 .....)([ f ( t ) ] 32200 ????? ???????? ? ????? f d tef d tef d tedttfe stststst)()()2()3( tftftftf ?????? ???163。)4(2)( s in22?? sst. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Laplace Transform Rule 5: if f (t + ω) = f (t), so that f (t) has period ω, then ??? ?? 0s )(e11[ f ( t ) ] dttfe st163。 sftf ?? )0()]([ 163。0)0( ?fstf ?)]([ 39。[cos(wt)] 22 wss??. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Laplace Transform Chapter 5 Laplace Transforms The Laplace transform of f(t) = sin2t ttf 2sin)( ? 則 ttttf 2s inc o ss in2)(39。 stf ?)]([2??? 163。 )]([ stf ? 163。1)0( ?f239。39。 ???t21 e)2s1(163。(y) = 2s1 163。 21 ?s2 163。(y) – s + 2 163。(y’’) = s2163。(y) = 163。 y(0) = 1, y’(0) = 2 163。[f (n)(t)] = sn 163。?w1?? {s 163。 })]wt[ c os (w1{ 39。[sin(wt)] = 163。 ?? 39。[f (t)]f (0) tMetf ??)(163。 asksask ekseeat ???? ???? 1lim)]([0?1)(0????dtat?. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Laplace Transform Rule 3: if for t ? t0, f(t) is continuous for t ? 0, and f ‘(t) is piecewise continuous on [0, k] for every k 0, then For s α 163。[f(t)] t f(t) t g(t) a Chapter 5 Laplace Transforms . Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Unit step function Chapter 5 Laplace Transforms . Hu, Assistant Professor, Department of Applied Phys
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