Laplace Transforms and Sage
John Travis
Mississippi College
%auto
t,s=var('t,s')
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# Find the laplace transform of f(t)=1
f(t)=1
assume(s>0)
F(s)=integrate(e^(-s*t)*f(t),t,0,oo)
html('$F(s) = %s$'%str(latex(F(s))))
(plot(f,(t,0,2),color='blue')+plot(F,(s,0,2),ymax=20,color='green')).show()
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# Find the laplace transform of f(t)=t
f(t)=t
assume(s>0)
F(s)=integrate(e^(-s*t)*f(t),t,0,oo)
html('$F(s) = %s$'%str(latex(F(s))))
(plot(f,(t,0,2),color='blue')+plot(F,(s,0,2),ymax=20,color='green')).show()
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# Find the laplace transform of f(t)=t^n
assume(s>0)
L2 = integrate(e^(-s*t)*t^2,t,0,oo)
L3 = integrate(e^(-s*t)*t^3,t,0,oo)
L4 = integrate(e^(-s*t)*t^4,t,0,oo)
L5 = integrate(e^(-s*t)*t^5,t,0,oo)
L6 = integrate(e^(-s*t)*t^6,t,0,oo)
L7 = integrate(e^(-s*t)*t^7,t,0,oo)
G = Graphics()
assume(s>0)
for n in range(2,8):
f(t)=t^n
F(s)=integrate(e^(-s*t)*f(t),t,0,oo)
G += plot(f,(t,0,2.5),color='blue')+plot(F,(s,0,3),ymax=20,color='green')
html.table([[t^2,t^3,t^4,t^5,t^6,t^7,' '],[L2,L3,L4,L5,L6,L7,' '],[' ',' ',' ',' ',' ',' ',G]])
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var('n')
assume(n,'integer')
assume(n>1)
laplace(t^n,t,s).show()
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var('b')
S = sin(b*t)
C = cos(b*t)
LS = laplace(S,t,s)
LC = laplace(C,t,s)
f(t)=sin(t)
assume(s>0)
F(s)=integrate(e^(-s*t)*f(t),t,0,oo)
G = plot(f,(t,0,2*pi),color='blue')+plot(F,(s,0,2*pi),ymax=2,color='green')
f(t)=cos(t)
assume(s>0)
F(s)=integrate(e^(-s*t)*f(t),t,0,oo)
H = plot(f,(t,0,2*pi),color='blue')+plot(F,(s,0,2*pi),ymax=2,color='green')
html.table([[S,C],[LS,LC],[G,H]])
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