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Coursework on Laplace Transforms Q1. Laplace Transforms is an integral transforms used to transform a real valued variable t (t?0) into a complex valued phenomenon s. The function of t say f(t) is transformed into a function of s say F(s) such that the functional relationship is written as L[f(t)]=F(s) where f(t) is of exponential order and the limit for f(t) exists.

The Laplace transforms formula is given below: L[f(t)]= here f(t) is piecewise continuous and the derivative of f(t) should exist in the domain of t (t?0) Q2. L[]==, s>k. Q3. The following are the Laplace transforms for the given S.No. Function y(t) Laplace Transform L{y(t)} Region of Convergence (ROC) 1 y(t) Y(s) n/a 2 a (=a constant) a/s Re(s)>0 3 t 1/s2 Re(s)>0 4 t2 2/s3 Re(s)>0 5 tn[n>0 n?N] [n(n–1)(n–2)….1]/sn+1 n!/sn+1 Re(s)>0 n>–1 6 e–at 1/(s+a) s>a 7 e–attn n!/(s+a)n+1 Re(s)>0 8 H(t–a) e–as/s Re(s)>0 9 ?(t) 1 For all s 10 ?(t–a) 1 For all s 11 cos(?t) s/(s2+?2) Re(s)>0 12 sin(?t) ?/(s2+?2) Re(s)>0 13 e–at cos(?t) (s+a)/[(s+a)2+?2] Re(s)>0 14 e–at sin(?t) ?/[(s+a)2+?2] Re(s)>0 15 dy/dt sY(s)–y0 Y(s)=L[y(t)],y0=y(0) 16 d2y/dt2 s2Y(s)–sy0–y’(0) y’(0)=dy/dt|t=0 Q4. (i) 3/s–5/s2+12/s3 (ii) 10/(s+4)+7/(s–1/2)=10/(s+4)+14/(2s–1) (iii) 2 (iv) 4s/(s2+9)+12/(s2+4) (v) 36/[(s+1)2+9] (vi) sY(s)–f0 where Y(s)=L[y(t)] and y0=y(0) Q5. (i) ?(t)+6+4t (ii) e2t (iii) 2e–5t+sin3t (iv) 1/3(e2t–e–t) (using partial fractions) (v) et/2sin5t (using shifting theorem) (vi) e–3tcos(2t) {using shifting theorem} Q6. ...

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