chapter5laplacetransforms-全文預(yù)覽
【正文】 n factor and Q(s) has higher degree than P(s). ))s(Q )s(P(163。 ?? yy)()()()()()]0()0()[( 39。Subsidiary equation (輔助方程式 ) )()0()0()()( 39。39。 ts ??? s in1)1(221 ???163。 163。? ?? ???? ?/0s/2 s ine1 1[ f ( t ) ] t dte st163。)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。0)0( ?fstf ?)]([ 39。 stf ?)]([2??? 163。1)0( ?f239。 ???t21 e)2s1(163。 21 ?s2 163。(y’’) = s2163。 y(0) = 1, y’(0) = 2 163。?w1?? {s 163。[sin(wt)] = 163。[f (t)]f (0) tMetf ??)(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 Physics, National University of Kaohsiung Unit step function Chapter 5 Laplace Transforms . Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Dirac’s Delta function Chapter 5 Laplace Transforms ))](()([1)( katuatukatf k ??????以兩個單位階梯函數(shù)來表示 163。[f(t)] = F(s) 163。[f(t)] ?? ? ?? ?? ??0sx02/1st dxe2dtte 2{163。1[G(s)] 4. 163。[af(t)] = a163。(eiwt) = 163。[sin(wt)] 22 wsw??Chapter 5 Laplace Transforms 利用分部積分 . Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Laplace Transform Theorem : 1. 163。39。1[F(s)] = f(t) The Laplace transform of f(t) = t is 163。. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 5 Laplace Transforms Laplace Transform 1. 拉普拉斯轉(zhuǎn)換乃 算子演算法 (operational calculus),它將微積分演算變成代數(shù)演算 .(為特殊的傅立葉轉(zhuǎn)換 ) 2. 拉普拉斯轉(zhuǎn)換在工程上用於機(jī)械以及電力的驅(qū)動力問題 ,特別是當(dāng)驅(qū)動力為不連續(xù) ,脈衝或是正弦 ,餘弦及更複雜的周期性函數(shù) . 3. 拉普拉斯轉(zhuǎn)換可直接解問題 ,求解初值問題時無需先求通解 ,且解非齊次微分方程時亦無需先求對應(yīng)之齊次方程式之解 . 4. 偏微分方程式也能以拉普拉斯轉(zhuǎn)換處理 . . Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 5 Laplace Transforms Laplace Transform The Laplace transform 163。[f(t)] = F(s) 163。39。[cos(wt)] 22 wss??同理可證 163。[g(t)] Whenever all three Laplace transform exist 163。(sinwt) 2222 wswiwss????2. For any real number a, 163。1[F(s)]+ 163。[f(t)] ?? ???02/1st dtteGamma function : ??????01xt dtte)x(163。[eatf(t)] = F(sa) For s a 163。[g(t)] = eas 163。[f ’(t)] = s 163。)]wt[ c os (w 1)wts in ( ??163。? })]wt{ [ c os (w1 39。[f (t)]sn1f (0)sn2f ’(0) …….. f (n1)(0) Solve the initial value problem : y’’ 4y = t 。(t) By rule 4 : 163。(t) = t)s1(163。(y) =
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