Mathematic Methods HW Solutions 44

Mathematic Methods HW Solutions 44 - Chapter 8 10.3 10.5...

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Unformatted text preview: Chapter 8 10.3 10.5 10.7 10.8 10.9 10.11 10.13 10.15 44 1 2 t sinh t 10.4 b(b − a)te−bt + a[e−bt − e−at ] 10.6 (b − a)2 −at a cosh bt − b sinh bt − ae a2 − b 2 e−at e−bt e−ct + + (b − a)(c − a) (a − b)(c − b) (a − c)(b − c) (2t2 − 2t + 1 − e−2t )/4 10.10 (1 − cos at − 1 at sin at)/a4 2 1 cos bt cos at 1 cos at − cos bt 10.12 2 − + 22 b 2 − a2 b − a2 b2 a2 ab (e−t + sin t − cos t)/2 10.14 e−3t + (t − 1)e−2t 10.17 y = t − 2, t > 2 0, t<2 11.8 y = e−2(t−t0 ) sin(t − t0 ), 0, 11.9 y = 11.10 y = 11.11 y = 1 −(t−t0 ) 3e 0, 1 3 0, (cosh at − 1)/a2 , t > 0 0, t<0 11.7 y = 1 3t 1 1 e + e−4t − et 14 35 10 11.1 y = 11.13 11.15 11.21 11.23 e−at + e−bt [(a − b)t − 1] (b − a)2 e−at − cosh bt + (a/b) sinh bt a2 − b 2 (t − t0 )e−(t−t0 ) , t > t0 0, t < t0 t > t0 t < t0 sin 3(t − t0 ), t > t0 t < t0 sinh 3(t − t0 ), t > t0 t < t0 1 2 [sinh(t 0, − t0 ) − sin(t − t0 )], t > t0 t < t0 (a) (a) (a) (a) 5δ (x − 2) + 3δ (x + 7) (b) 3δ (x + 5) − 4δ (x − 10) 1 (b) 0 (c) −3 (d) cosh 1 8 (b) φ(|a|)/(2|a|) √ (c) 1/2 (d) 1 π δ (x + 5)δ (y − 5)δ (z ), δ (r − 5 2 )δ (θ − 34 )δ (z )/r, √ π δ (r − 5 2 )δ (θ − π )δ (φ − 34 )/(r sin θ) 2 π (b) δ (x)δ (y + 1)δ (z + 1), δ (r − 1)δ (θ − 32 )δ (z + 1)/r, √ π π δ (r − 2 )δ (θ − 34 )δ (φ − 32 )/(r sin θ) √ √ (c) δ (x + 2)δ (y )δ (z − 2 3 ), δ (r − 2)δ (θ − π )δ (z − 2 3 )/r, δ (r − 4)δ (θ − π )δ (φ − π )/(r sin θ) 6 √ √ √ π (d) δ (x − 3)δ (y + 3)δ (z + 6 ), δ (r − 3 2 )δ (θ − 74 )δ (z + 6 )/r, √ π π δ (r − 2 6 )δ (θ − 23 )δ (φ − 74 )/(r sin θ) 11.25 (a) and (b) F (x) = δ (x) − 2δ (x) (c) G (x) = δ (x) + 5δ (x) 12.2 y = sin ωt − ωt cos ωt 2ω 2 12.6 y = (cosh at − 1)/a2 , t > 0 12.8 y = 12.11 y = 1 − e−t − te−t , 0 < t < a (t + 1 − a)ea−t − (t + 1)e−t , 1 −3 sin 2x sin ωt − ω cos ωt + ωe−t ω (1 + ω 2 ) a(cosh at − e−t )− sinh at 12.7 y = a(a2 −1) 12.3 y = t>a 12.12 y = cos x ln cos x + (x − π ) sin x 2 ...
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