6.2 Introduction-to-Partial-Differential-Equations-Third-Edition-by-K-Sankara-Rao

324 INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS (iv) L[eattn ; s] n! n! sn1 so(sa) (s  a)n1 (v) L[cos at cosh bt; s] ª at § ebt  ebt · º L «cos ¨© 2 ¸¹ ; s» ¼ ¬ 1 {L[ebt cos at; s]  L[ebt cos at; s]} 2 1 ­ sb  sb ½ 2 ®  b)2   b)2  ¾ ¯ (s a2 (s a2 ¿ EXAMPLE 6.6 Fing the Laplace transform of the following: (i) t2eat , (ii) t sin at, (iii) t2 cos at, (iv) tneat . Solution Using the result established in Theorem 6.4, we have (i) L[t2eat ; s] (1)2 d2 L[eat ; s] d2 § 1 · d § 1 · 2 ds2 ds2 ¨© s  a ¸¹ ds ©¨ (s  a)2 ¹¸ (s  a)3 Alternatively, L[eatt2; s] 2! 2 (using the shifting property) s3 so(sa) (s  a)3 (ii) L[t sin at; s] (1)1 d L[sin at; s]  d § s2 a a2 · 2as ds ds ¨©  ¹¸ (s2  a2 )2 (iii) L[t2 cos at; s] (1)2 d2 L[cos at; s] d2 § s · ds2 ds2 ©¨ s2  a2 ¹¸ d ª a2  s2 º 2s3  6sa2 « » ds ¬« (s2  a2 )2 ¼» (s2  a2 )3 (iv) L[eattn ; s] n! n! (using the shifting property) sn1 so(sa) (s  a)n1 EXAMPLE 6.7 Find the Laplace transform of (i) te4t sin 3t, (ii) sin 2t sin 3t, (iii) sin3 2t. www.MathSchoolinternational.com