DO # 8 We need to assume that s>2 (from 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 ) in order for the improper integral to converge. assume(s>2): interface(showassumed=0): int(exp(-s*t)*exp(2*t)*cos(3*t),t=0..infinity): simplify(%);

KiYsJkkjc3xpckc2IiIiIiIiIyEiIkYmLCgqJClGJEYnRiZGJiomIiIlRiZGJEYmRigiIzhGJkYo

DO # 11 The laplace transform does not deal directly with piecewiuse continuous functions: we have to use convert( ,Heaviside). restart: assume(s>0): interface(showassumed=0): f:=piecewise(t>0 and t<3,exp(2*t),t>3,1);

PiUiZkctJSpQSUVDRVdJU0VHNiQlIj9HRic=

with(inttrans): convert(f,Heaviside); laplace(%,t,s);

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LCYqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiJCIiIkkjc3xpckdGKUYuISIiRi5GL0YwRi4qJiwmRi9GLiIiI0YwRjAsJkYuRi4tRiU2IywmIiInRi5GLEYwRjBGLkYu

Notice that the laplace transform is "well--defined" in s=2.

DO # 10 Multiplication by t corresponds to taking one derivative with respect to s and multiplying by -1. Multiplying by exp(at) translates the Laplace transform right by a, i.e., s is replaced by s-a. In this case a=-1. restart: with(inttrans): laplace(sin(2*t),t,s);

LCQqJiIiIyIiIiwmKiQpSSJzRzYiRiRGJUYlIiIlRiUhIiJGJQ==

(-1)*diff(%,s); subs(s=s+1,%);

LCQqKCIiJSIiIiksJiokKUkic0c2IiIiI0YlRiVGJEYlRiwhIiJGKkYlRiU=

LCQqKCIiJSIiIiksJiokKSwmSSJzRzYiRiVGJUYlIiIjRiVGJUYkRiVGLSEiIkYqRiVGJQ==

Comparing laplace(exp(-t)*t*sin(2*t),t,s); It is the same as before as "I" in MAPLE is the imaginary unit "i".

LCQqKiIiJSIiIiwmSSJzRzYiRiVGJUYlRiUpLCZGJ0YlLCZGJUYlKiYiIiNGJV4jRiVGJSEiIkYlRi1GLyksJkYnRiUsJkYlRiVGLEYlRiVGLUYvRiU=

DO # 17 To take the Laplace transform of a product of trig functions, we can use double angle and sum/difference formulas. These are accessed through Maple's combine( ,trig) command. restart: with(inttrans): sin(3*t)*cos(3*t); combine(%,trig);

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LCQqJiMiIiIiIiNGJS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiJ0YlSSJ0R0YsRiVGJUYlRiU=

laplace(%,t,s);

LCQqJiIiJCIiIiwmKiQpSSJzRzYiIiIjRiVGJSIjT0YlISIiRiU=

Comparing laplace(sin(3*t)*cos(3*t),t,s);

LCQqJiIiJCIiIiwmKiQpSSJzRzYiIiIjRiVGJSIjT0YlISIiRiU=

DO # 33 restart: assume(s>0): interface(showassumed=0): f:=piecewise(t>0 and t<2,0,t>2,1);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkqbXZlcmJhdGltR0YkNiNRJSUiZkdGJy1JI21vR0YkNi1RIzo9RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctRiw2I1E4LSUqUElFQ0VXSVNFRzYkJSI/RyUiP0dGJw==

with(inttrans): convert(f,Heaviside); laplace(%,t,s);

LUkqSGVhdmlzaWRlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJiIiIyEiIkkidEdGJyIiIg==

KiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiYiIiMiIiJJI3N8aXJHRihGLSEiIkYtRi5GLw==

comparing laplace(1,t,s);

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It is precisely the laplace transform of f(t)=1(which is 1/s), times exp(-2s).