John Travis
Mississippi College
var('t,s')
# f=t^30
f = e^(4*t+8)
L=laplace(f,t,s)
html.table([['$f(t) =$',f],['$F(s) =$',L]])
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F = 14/((3*s+2)*(s-4))
IL=inverse_laplace(F,s,t)
html.table([['$F(s) =$',F],['$f(t) =$',IL]])
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# If one wants to do this inverse transform by hand, then partial fractions may need to be used
# to break up the term into linear denominators
F.partial_fraction(s).show()
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# Using these with differential equations
var('t,s,y,y0')
y = function('y',t)
eqn = diff(y,t) - 5*y == e^(-t)
# eqn = diff(y,t) + 4*y == 8*sin(4*t)
ICS =[0,y0]
print 'Differential Equation:'
eqn.show()
print 'With initial condition',ICS
desolve_laplace(eqn,y,ICS).show()
Differential Equation: With initial condition [0, y0] Differential Equation: With initial condition [0, y0] |
# to quickly solve (or to check your manual solution)
var('t,y,C')
y=function('y',t)
@interact
def _(eq = input_box(diff(y,t) + 5*y == exp(-t),label='DE:'),y0 = input_box(2,width=2,label='$y_0$')):
html('Remember to enter diff(y,t) for any derivative.')
ics = [0,y0]
eq.show()
desolve_laplace(eq,y,ics).show()
Click to the left again to hide and once more to show the dynamic interactive window |
# Notice the Sage commands can handle the translation theorem forward
# but cannot deal with the translation theorem in reverse
var('t')
u(t) = unit_step(t)
F = laplace(u(t-3),t,s)
F.show()
f = inverse_laplace(F,s,t)
f.show()
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# We can have Sage help with parts along the way and then apply the theorems ourselves
# Below, consider e^(-2s)*F(s)
var('t,s')
F = 4/(s*(s+3))
f =inverse_laplace(F,s,t)
f(t = t-2)*u(t-2)
-4/3*(e^(-3*t + 6) - 1)*unit_step(t - 2) -4/3*(e^(-3*t + 6) - 1)*unit_step(t - 2) |
Sandbox below...
var('t,s')
F = (1)/((s+5)*(s-2))
IL=inverse_laplace(F,s,t)
html.table([['$F(s) =$',F],['$f(t) =$',IL]])
F.partial_fraction(s).show()
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# periodic function
var('t')
f1 = 2*unit_step(t-2)
f2 = 1*unit_step(t-3)
f3 = -3*unit_step(t-5)
f4 = 2*unit_step(t-7)
f5 = 1*unit_step(t-8)
f6 = -3*unit_step(t-10)
f = f1+f2+f3+f4+f5+f6
G = plot(0,t,0,2,thickness = 5,ticks=[[1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4]])+plot(2,t,2,3,thickness = 5)+plot(3,t,3,5,thickness = 5)+plot(0,t,5,7,thickness = 5)+plot(2,t,7,8,thickness = 5)+plot(3,t,8,10,thickness = 5)
G += circle((10.5,1.5),.1,fill=true)+circle((11,1.5),.1,fill=true)+circle((11.5,1.5),.1,fill=true)
G.show()
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