7-5-S-Laplace Transform Issues and Opportunities in Mathematica

Transcription

1 7-5-S-Laplace Tranform Iue and Opportunitie in Mathematica The Laplace Tranform i a mathematical contruct that ha proven very ueful in both olving and undertanding differential equation. We introduce it and how it power here. Definition of Laplace Tranform of a function f(t) if the integral converge: L (f (t)) () = F () = 0 e - t f (t) dt In order to evaluate uch an integral we might be uing integration by part for f(t) of a different "ort" than exponential function. However, we turn to Mathematica to do the integration for u and ue it own powerful computation, look-up, and pattern matching algorithm to do Laplace Tranform. Thi "trange" integral tranform mot function (if the integral converge) from a t domain to an domain where i (in our cae) a real poitive number. Integrate[Exp[- t] f[t], {t, 0, Infinity}] ConditionalExpreion, Re[] > 0 Integrate[Exp[- t] Co[3 t], {t, 0, Infinity}] ConditionalExpreion, Re[] > Notice the retriction that Re[]>0 in order for u to have convergence to the quantity, /(9+^2). Thu /(9+^2) IS the Laplace Tranform of the function co(3t). Table of thee exit and we could viit them, however ince Mathematica can do thee intantly without u looking them up we do our computation of Laplace Tranform in Mathematica.

2 2 9-5-S-OverViewLaplaceTranform-StudentVerion.nb Often the domain i called the frequency domain, and the t domain i called the time domain, for if the following integral - the Laplace Tranform - i to make ene with repect to unit then the unit of mut be /Time, for in thi way *t ha no unit and o Exp[-*t] ha no unit. Thu the Laplace Tranform integral i obtained by jut integrating a function of t over an interval of time, [0, ) - reulting in a function of, we call it L (f(t))() = F(). F[_] = Integrate[Exp[- t] f[t], {t, 0, Infinity}] e - t f[t] dt 0 Mathematica can compute thee Laplace Tranform and we ee how to do thi. Here are the definition brought up by query from Mathematica. If one need more, uch a an example, then go to Help Menu and type "LaplaceTranform" and you will have a window offering up a definition with an example worked out at the bottom.or jut type one of the following?? LaplaceTranform LaplaceTranform[expr, t, ] give the Laplace tranform of expr. LaplaceTranform[expr, {t, t 2, }, {, 2, }] give the multidimenional Laplace tranform of expr. Attribute[LaplaceTranform] = {Protected, ReadProtected} Option[LaplaceTranform] = {Aumption $Aumption, GenerateCondition Fale, PrincipalValue Fale, Analytic True} We have a tougher trick in undoing the Laplace Tranform and thi i done uing the InvereLaplace- Tranform command.?? InvereLaplaceTranform InvereLaplaceTranform[expr,, t] give the invere Laplace tranform of expr. InvereLaplaceTranform[expr, {, 2, }, {t, t 2, }] give the multidimenional invere Laplace tranform of expr. Attribute[InvereLaplaceTranform] = {Protected, ReadProtected}

3 9-5-S-OverViewLaplaceTranform-StudentVerion.nb 3 So here we go... taking Laplace Tranform of function.... We Build Up a Table of Laplace Tranform and attempt to acertain pattern. LaplaceTranform[, t, ] LaplaceTranform[t, t, ] 2 LaplaceTranform[t^ 2, t, ] 2 3 LaplaceTranform[t^ 3, t, ] 6 4 LaplaceTranform[t^ 4, t, ] 24 5 What do you believe we might get for the Laplace Tranform of t n? Of coure, we could top and offer a formal induction proof of general formula, but we proceed to take the Laplace Tranform of other clae of function. Trig function LaplaceTranform[Sin[3 t], t, ] LaplaceTranform[Sin[5 t], t, ] LaplaceTranform[Co[3 t], t, ] LaplaceTranform[Co[5 t], t, ] See a pattern here?

4 4 9-5-S-OverViewLaplaceTranform-StudentVerion.nb... and exponential function. LaplaceTranform[Exp[3 t], t, ] -3 + LaplaceTranform[Exp[5 t], t, ] -5 + LaplaceTranform[Exp[- 7 t], t, ] 7 + Incidentally, the tranform of function like f(t) = e a t and f(t) = k are eay by hand effort! Try one. Continuing in thi manner of dicovery we could fill in a typical Laplace Tranform Table. A ample, but hort and imple form, table i the following: We could combine trig and exponential function.. LaplaceTranform[Exp[5 t] Sin[3 t], t, ] (-5 + ) 2... and in general to match the table entry.

5 9-5-S-OverViewLaplaceTranform-StudentVerion.nb 5 LaplaceTranform[Exp[a t] Sin[b t], t, ] b b 2 + (a - ) 2 If we are given the Laplace Tranform Y() how might we obtain the original, i.e. the y(t) whoe Laplace Tranform i Y(). We try it on ome of our oberved Tranform and ome other function. InvereLaplaceTranform ( - 4),, t e 4 t InvereLaplaceTranform ^2 + 4,, t 3 4 i e(3-2 i) t - + e 4 i t + DiracDelta [t] InvereLaplaceTranform ^7,, t t InvereLaplaceTranform ^ ( - 4),, t e 4 t + Co[2 t] InvereLaplaceTranform ^2-4,, t 2 e-2 t + e 4 t yol[t_] = InvereLaplaceTranform ^2 + 4,, t 2 e(-3-2 i) t + e 4 i t Plot[yol[t], {t, 0, 2}]

6 6 9-5-S-OverViewLaplaceTranform-StudentVerion.nb InvereLaplaceTranform ^2 + 4,, t 2 e(-3-2 i) t + e 4 i t Here we need to ue Euler' formula, e i θ = co(θ) + i in(θ), invoked by Mathematica Complex Expand command to offer a real preentation of our tranform. ComplexExpand InvereLaplaceTranform ^2 + 4,, t 2 e-3 t Co[2 t] + 2 e-3 t Co[2 t] Co[4 t] + 2 e-3 t Sin[2 t] Sin[4 t] + i - 2 e-3 t Sin[2 t] - 2 e-3 t Co[4 t] Sin[2 t] + 2 e-3 t Co[2 t] Sin[4 t] Further, we invoke Mathematica' TrigReduce command to ue trigonometric identitie to reduce our Invere Laplace Tranform to a imple form from which we could ee o much, in thi cae a damped ocillator! TrigReduce ComplexExpand InvereLaplaceTranform ^2 + 4,, t e -3 t Co[2 t] Let u take the Laplace Tranform of general function and derivative.... LaplaceTranform[y[t], t, ] LaplaceTranform[y[t], t, ] LaplaceTranform[y'[t], t, ] LaplaceTranform[y[t], t, ] - y[0] We take Laplace Tranform of econd derivative... LaplaceTranform[y''[t], t, ] 2 LaplaceTranform[y[t], t, ] - y[0] - y [0] We take Laplace Tranform of third derivative... LaplaceTranform[y'''[t], t, ] 3 LaplaceTranform[y[t], t, ] - 2 y[0] - y [0] - y [0] We take Laplace Tranform of fourth derivative... LaplaceTranform[y''''[t], t, ] 4 LaplaceTranform[y[t], t, ] - 3 y[0] - 2 y [0] - y [0] - y (3) [0]

7 9-5-S-OverViewLaplaceTranform-StudentVerion.nb 7 Let u give LaplaceTranform[y[t],t,] a name, ay Y[]. Then thing look nicer. LaplaceTranform[y'[t], t, ] /. {LaplaceTranform[y[t], t, ] Y[]} -y[0] + Y[] LaplaceTranform[y''[t], t, ] /. {LaplaceTranform[y[t], t, ] Y[]} - y[0] + 2 Y[] - y [0] Can we take the Laplace Tranform of a whole differential equation? Let' try. eq = LaplaceTranform[a y''[t] + b y'[t] + c y[t] Sin[t], t, ] c LaplaceTranform[y[t], t, ] + b ( LaplaceTranform[y[t], t, ] - y[0]) + a 2 LaplaceTranform[y[t], t, ] - y[0] - y [0] + 2 Let u try to replace LaplaceTranform[y[t],t,] with Y[] here too. eq = LaplaceTranform[a y''[t] + b y'[t] + c y[t] Sin[t], t, ] /. {LaplaceTranform[y[t], t, ] Y[]} c Y[] + b (-y[0] + Y[]) + a - y[0] + 2 Y[] - y [0] + 2 We can olve for the Laplace of the function in the above differential equation. YSol[_] = Y[] /. Solve[eq, Y[]][[]] + b y[0] + a y[0] + b 2 y[0] + a 3 y[0] + a y [0] + a 2 y [0] + 2 c + b + a 2 Now if we could compute the Invere Laplace Tranform of Y () we would have the olution to our differential equation, y (t). Let u do thi. Our olution appear a bit on the wild ide involving the contant a, b, and c, and the all too familiar radical b 2-4 ac one obtain from the characteritic equation of the homogeneou portion of our original differential equation a y (t) + b y (t) + c y(t) = in(t) in uing eigenvalue olution trategie.

8 8 9-5-S-OverViewLaplaceTranform-StudentVerion.nb InvereLaplaceTranform[YSol[],, t] // Simplify e - b+ b2-4 a c t 2 a -2 a 2 - b a c + b b 2-4 a c + 2 a 2 e b2-4 a c t a + b 2 e b2-4 a c t a - 2 a c e b2-4 a c t a + b b 2-4 a c e b2-4 a c t b+ b2-4 a c t a - 2 b b 2-4 a c e 2 a Co[t] - 2 (a - c) b 2-4 a c e b+ b2-4 a c t 2 a Sin[t] - a 2 b y[0] - b 3 y[0] + 2 a b c y[0] - b c 2 y[0] + a 2 b 2-4 a c y[0] + b 2 b 2-4 a c y[0] - 2 a c b 2-4 a c y[0] + c 2 b 2-4 a c y[0] + a 2 b e b2-4 a c t a y[0] + b 3 e b2-4 a c t a y[0] - 2 a b c e b2-4 a c t a y[0] + b c 2 e b2-4 a c t a y[0] + a 2 b 2-4 a c e b2-4 a c t a y[0] + b 2 b 2-4 a c e b2-4 a c t a y[0] - 2 a c b 2-4 a c e b2-4 a c t a y[0] + c 2 b 2-4 a c e b2-4 a c t a y[0] - 2 a 3 y [0] - 2 a b 2 y [0] + 4 a 2 c y [0] - 2 a c 2 y [0] + 2 a 3 e b2-4 a c t a y [0] + 2 a b 2 e b2-4 a c t a y [0] - 4 a 2 c e b2-4 a c t a y [0] + 2 a c 2 e b2-4 a c t a y [0] 2 b 2-4 a c a 2 + b 2-2 a c + c 2 The Tranfer function of a differential equation uch a a*y (t) + b*y (t) + c*y(t) = in(t) i Q() = / (a* 2 + b* + c) ( over the denominator in the Laplace Tranform of the olution) and notice the coefficient a, b, and c, of the LHS or of the ytem - not the driver in(t). Thi characterize the ytem repone and the root of the characteritic equation, a* 2 + b* + c = 0, are the characteritic root or eigenvalue of thi ytem. Engineer get to "know" ytem and the ytem' behavior by their Tranfer function and the eigenvalue; electrical engineer call the eigenvalue "pole." Indeed, if we let y(0) = 0 and y'(0) = 0 then YSol[] = L[]*Q[], and o Q[] = Yol[]/L[], where YSol[] i the output' Laplace Tranform and L[] i the input' Laplace Tranform. Thi effectively how that the Laplace Tranform of the Solution i imply a multiplication of Laplace Tranform of the input or driver function, i.e. L[], and the tranfer function Q[] = Q() = / (a* 2 + b* + c). The latter contain all the action of the differential equation model term a*y (t) + b*y (t) + c y(t) while the former, L[], contain the action of the driver function, in thi cae in(t).

9 9-5-S-OverViewLaplaceTranform-StudentVerion.nb 9 So when we actually ue reaonable initial condition (y(0) = 0, no initial diplacement, and y (0) = 0, no initial velocity - depending upon the driver to get ome activity going) on the invere Laplace Tranform, YSol[], of our differential equation a*y (t) + b*y (t) + c*y(t) = in(t) YSol[] /. {y[0] 0, y'[0] 0} + 2 c + b + a 2 We try a pecific example, one we know we can olve in everal other way - by hand, with Mathematica DSolve. We conider y''(t) + 6 y'(t) + 5 y(t) = in(t), y(0) = 4, y'(0) = 0. Firt we ue DSolve and ome formatting to grab the olution for comparion. yol[t_] = y[t] /. DSolve[{ y''[t] + 6 y'[t] + 5 y[t] Sin[t], y[0] 4, y'[0] 0}, y[t], t][[ ]] // Expand e-5 t e-t Co[t] + Sin[t] We conider thi ame differential equation and apply Laplace Tranform to all term, i.e. both ide of the differential equation. eq = LaplaceTranform[ y''[t] + 6 y'[t] + 5 y[t] Sin[t], t, ] /. {LaplaceTranform[y[t], t, ] Y[], y[0] 4, y'[0] 0} Y[] + 2 Y[] + 6 (-4 + Y[]) + 2 Now we olve for Y[], the Laplace Tranform of y[t] in order to get ready for the invere Laplace Tranform. L[_] = Y[] /. Solve[eq, Y[]][[]] Now we apply the invere tranform to get the olution for y[t]. l[t_] = InvereLaplaceTranform[L[],, t] // Expand e-5 t e-t Co[t] + Sin[t] Notice the two portion of thi oloution, namely the Tranient olution e-5 t + 4 e-t 8 and the Teady State - 3 Co[t] 26 + Sin[t]. Thee would be 3

10 0 9-5-S-OverViewLaplaceTranform-StudentVerion.nb produced by variou hand technique or Mathematica DSolve approach which we how here. How doe that compare to the DSolve olution? Exactly the ame a we ee when, again, we ue DSolve to obtain a olution. yol[t_] = y[t] /. DSolve[{ y''[t] + 6 y'[t] + 5 y[t] Sin[t], y[0] 4, y'[0] 0}, y[t], t][[ ]] // Expand e-5 t e-t Co[t] + Sin[t] So what i the big deal? Who would omeone chooe Laplace Tranform to olve a problem if DSolve or by hand can do it directly? Here i the point (well one of the many point of advantage) of Laplace Tranform. They convert calculu to algebra, then we play in algebra land, and then we return to calculu land. More formally, the Laplace Tranform tranform the problem from the time domain (t) to the frequency domain in (). We conider two additional "Real Good" reaon for Laplace Tranform Unit Step or Heaviide Function - what doe thi function do. f[x_] = UnitStep[x] UnitStep[x]

11 9-5-S-OverViewLaplaceTranform-StudentVerion.nb Plot[f[x], {x, -8, 8}, PlotStyle {Thickne[.0]}] f2[x_] = UnitStep[x - 3] UnitStep[-3 + x] Plot[f2[x], {x, -8, 8}, PlotStyle {Thickne[.0]}] We could define our own unit tep function with conditional or if tatement a follow, but Mathematica already ha it defined o we accept UnitStep[x] in Mathematica. unittep(x) =, if x>0, and 0 otherwie. unittep[x_] = If[x > 0,, 0] If[x > 0,, 0]

12 2 9-5-S-OverViewLaplaceTranform-StudentVerion.nb Plot[unittep[x], {x, - 8, 8}, PlotStyle {Thickne[.0]}] Dirac Delta or Impule Function g[x_] = DiracDelta[x] DiracDelta[x] Strange plot.... Plot[g[x], {x, -4, 4}, PlotStyle {Thickne[.0]}] We examine the Dirac Delta function ome more. Thi function repreent an intantaneou change. E.g., we change the alt concentration by "dumping" in more alt ALL AT ONCE! OR we "hock" a pring ma giving it an intantaneou force ALL AT ONCE! The integral of an interval containing x = 0 for the DiracDelta[x] function i. What thi ay i the impule offered by a driver of the form DiracDelta[x] impart force of ize unit to our ytem, BUT all at once like the quick bang of a hammer, rather than a gradual application of force. We check out ome propertie of the DiracDelta[x] function. Integrate[g[x], {x, -, }]

13 9-5-S-OverViewLaplaceTranform-StudentVerion.nb 3 Integrate[g[x], {x,.5, }] 0. g[.] 0 g[0] DiracDelta[0] g[2] 0 Tranform of UnitStep and Dirac Function - where doe thi fit into our Table of Laplace tranform? f[t_] = UnitStep[t]; LaplaceTranform[f[t], t, ] g[t_] = DiracDelta[t]; LaplaceTranform[g[t], t, ] UnitStep function repreent a "tep" or finite immediate change in omething. DiracDelta function repreent an intantaneou action at one point... zero elewhere. del[x_, a_] = 2 a (UnitStep[x - (-a)] - UnitStep[x - a]) -UnitStep[-a + x] + UnitStep[a + x] 2 a

14 4 9-5-S-OverViewLaplaceTranform-StudentVerion.nb Plot[del[x, ], {x, -8, 8}] Integrate[del[x, ], {x, - 00, 00}] We examine an animation or a Manipulation of del[x,a] a a get maller an maller. We ay DiracDelta[x] i the limit of del[x,a] a a 0. Here are the plot in Manipulation to illutrate thi point. By clicking on the + box in thi command one can ee the value of a and by uing the lider next to a one can ee the effect of decreaing the value of a in the plot of the reulting function. Manipulate[Plot[del[x, a], {x, - 8, 8}, PlotRange {{-8, 8}, {0, 0}}, PlotPoint 00], {a, 3,.00, -.000}] a

15 9-5-S-OverViewLaplaceTranform-StudentVerion.nb 5 We can combine UnitStep function to turn thing on and then off. Conider thi function. h[x_] = UnitStep[x - 3] - UnitStep[x - 0] -UnitStep[-0 + x] + UnitStep[-3 + x] Plot[h[x], {x, 0, 20}, PlotStyle {Thickne[.0]}] Suppoe we have a ma pring ytem and we want to ee the effect of adding additional weight (force) on the ma after it i in motion, ay adding 5 unit of force at time t = 2. eq = LaplaceTranform[ y''[t] + 6 y'[t] + 5 y[t] 5 UnitStep[t - 2], t, ] /. {LaplaceTranform[y[t], t, ] Y[], y[0] 4, y'[0] 0} Y[] + 2 Y[] + 6 (-4 + Y[]) 5 e-2 Now we olve for Y[], the Laplace Tranform of y[t] in order to get ready for the invere Laplace Tranform. L[_] = Y[] /. Solve[eq, Y[]][[]] e e e Now we apply the Invere Laplace Tranform to get the olution for y[t]. yl[t_] = InvereLaplaceTranform[L[],, t] 4 e-5 t e 4 t + e e 5 t - 5 e 2+4 t HeaviideTheta[-2 + t] From the plot we ee the change in the action at t = 2 due to our applying additional force at that time.

16 6 9-5-S-OverViewLaplaceTranform-StudentVerion.nb Plot[yl[t], {t, 0, 0}, PlotRange {{0, 0}, {0, 4}}] Suppoe we have a pring ma ytem and we want to ee the effect of hitting it with pike force of 4 unit of force after two (t = 2) econd of "uual" motion. eq = LaplaceTranform[ y''[t] + 6 y'[t] + 5 y[t] 4 DiracDelta[t - 2], t, ] /. {LaplaceTranform[y[t], t, ] Y[], y[0] 0, y'[0] 0} 5 Y[] + 6 Y[] + 2 Y[] 4 e -2 Now we olve for Y[], the Laplace Tranform of y[t] in order to get ready for the invere Laplace Tranform. L[_] = Y[] /. Solve[eq, Y[]][[]] 4 e Now we apply the tranform to get the olution for y[t]. yl[t_] = InvereLaplaceTranform[L[],, t] e -5 (-2+t) - + e 4 (-2+t) HeaviideTheta[-2 + t] From the plot we ee the change in the action at t = 2 due to our applying a pike force at that time. Up until the pring ma wa at ret due to it initial condition of y(0) = 0 and y (0) = 0.

17 9-5-S-OverViewLaplaceTranform-StudentVerion.nb 7 Plot[yl[t], {t, 0, 0}, PlotRange {{0, 0}, {0, }}] Let u conider a tank ytem in which we "dump" tuff (DiracDelta function) and we alter amount uddenly (UnitStep function). Firt the ytem without any dratic dumping. Can you draw a cenario for thi ytem given the equation? eq = x'[t] (4 * y[t]) (7 * x[t]) 200, y'[t] (7 * x[t]) (7 * y[t]) 200, x[0] 50, y[0] 0 x [t] - 7 x[t] y[t] 50, y [t] 7 x[t] y[t], x[0] 50, y[0] ol = DSolve[eq, {x[t], y[t]}, t] x[t] 25 e - 7 t t 00 + e 7 t 50, y[t] e - 7 t t e 7 t 50 x[t_] = x[t] /. ol[[]]; y[t_] = y[t] /. ol[[]]; Plot[{x[t], y[t]}, {t, 0, 00}, PlotRange {{0, 00}, {0, 50}}] Let u conider a tank ytem in which we () "dump" tuff (DiracDelta function),

18 8 9-5-S-OverViewLaplaceTranform-StudentVerion.nb (2) we alter amount uddenly (UnitStep) function, or combination of thee. What are we dumping into the ytem? Where are we dumping it in? When do we tart dumping? How often are we dumping it in? What doe thee tuff function do? tuff[t_] = 40 DiracDelta[t - 25]; tuff2[t_] = 40 UnitStep[t - 25]; tuff3[t_] = 40 UnitStep[t - 25] - 20 UnitStep[t - 30]; tuff4[t_] = Sum[20 DiracDelta[t - a], {a, 20, 200, 20}]; We might then modify the model ytem with the new "tuff" function and Laplace away! tuff[t_] = tuff[t] 40 DiracDelta[-25 + t] eq = x'[t] 4 y[t] x[t] tuff[t], y'[t] 7 x[t] y[t] 200, x[0] 50, y[0] 0 x [t] 40 DiracDelta[-25 + t] - 7 x[t] y[t] 50, y [t] 7 x[t] y[t], x[0] 50, y[0] eq[[]] x [t] 40 DiracDelta[-25 + t] - 7 x[t] y[t] 50 Now ue LaplaceTranform yntax, olve, plot, and examine olution. eq = LaplaceTranform[eq[[]], t, ] /. {LaplaceTranform[x[t], t, ] X[], LaplaceTranform[y[t], t, ] Y[], x[0] 50, y[0] -> 0} X[] 40 e X[] Y[] 50 eq2 = LaplaceTranform[eq[[2]], t, ] /. {LaplaceTranform[x[t], t, ] X[], LaplaceTranform[y[t], t, ] Y[], x[0] 50, y[0] -> 0} Y[] 7 X[] Y[] 200

19 9-5-S-OverViewLaplaceTranform-StudentVerion.nb 9 XYol = Solve[{eq, eq2}, {X[], Y[]}] X[] 2000 e e 25 ( ) , Y[] e e Xol[_] = X[] /. XYol[[]] // Expand e e Yol[_] = Y[] /. XYol[[]] // Expand e xol[t_] = InvereLaplaceTranform[Xol[],, t] 5 5 e t + e 7 t e (-25+t) + e 50 7 (-25+t) HeaviideTheta[-25 + t] yol[t_] = InvereLaplaceTranform[Yol[],, t] e t - + e 7 t 50-4 e (-25+t) - + e 50 7 (-25+t) HeaviideTheta[-25 + t] After olving uing Laplace Tranform technique we plot our olution for x(t). We ee that from the dumping due to tuff[t] = 40*DiracDelta[t-25] of 40 additional unit intantly at time t = 25 there i a jump in x(t) at time t = 25 and an attendant rapid change in the econd tank, y(t). xplot = Plot[xol[t], {t, 0, 200}, PlotRange {{0, 200}, {0, 00}}]

20 S-OverViewLaplaceTranform-StudentVerion.nb yplot = 00 Plot[yol[t], {t, 0, 200}, PlotRange {{0, 200}, {0, 00}}, PlotStyle Hue[.3]] Here we ee both x (t) in gold and y (t) in green howing the effect of our dump in both tank. Show[xPlot, yplot]

21 9-5-S-OverViewLaplaceTranform-StudentVerion.nb 2 Suppoe we have a pring ma ytem and we want to ee the effect of turning on a force and turning off that force, over and over again, perhap by placing the ma in an alternating magnetic field. We model thi with mall rectangular function of force, 0.5 unit high and unit long, paced 2 unit apart, centered at, 3, 5, etc. rut[t_] = Sum 5 UnitStep[t - 2 k] - 5 UnitStep t - 2 k +, {k, 0, 0} -5 UnitStep[-2 + t] + 5 UnitStep[-20 + t] - 5 UnitStep[-9 + t] + 5 UnitStep[-8 + t] - 5 UnitStep[-7 + t] + 5 UnitStep[-6 + t] - 5 UnitStep[-5 + t] + 5 UnitStep[-4 + t] - 5 UnitStep[-3 + t] + 5 UnitStep[-2 + t] - 5 UnitStep[- + t] + 5 UnitStep[-0 + t] - 5 UnitStep[-9 + t] + 5 UnitStep[-8 + t] - 5 UnitStep[-7 + t] + 5 UnitStep[-6 + t] - 5 UnitStep[-5 + t] + 5 UnitStep[-4 + t] - 5 UnitStep[-3 + t] + 5 UnitStep[-2 + t] - 5 UnitStep[- + t] + 5 UnitStep[t] prut = Plot[rut[t], {t, 0, 20}, PlotRange {{0, 20}, {0, 6}}] We apply our force function rut(t). eq = LaplaceTranform[ y''[t] + 6 y'[t] + 5 y[t] rut[t], t, ] /. {LaplaceTranform[y[t], t, ] Y[], y[0] 0, y'[0] 0} 5 Y[] + 6 Y[] + 2 Y[] 5-5 e-2 5 e -7 5 e e e-8-5 e-5-5 e e e e-20-5 e-3-5 e e-4-5 e e-2-5 e e-8-5 e- + 5 e e-0-5 e- Now we olve for Y[], the Laplace Tranform of y[t] in order to get ready for the invere Laplace Tranform. -

22 S-OverViewLaplaceTranform-StudentVerion.nb L[_] = Y[] /. Solve[eq, Y[]][[]] e e - e 2 + e 3 - e 4 + e 5 - e 6 + e 7 - e 8 + e 9 - e 0 + e - e 2 + e 3 - e 4 + e 5 - e 6 + e 7 - e 8 + e 9 - e 20 + e 2 We finally apply the Invere Laplace Tranform to get the olution for y[t]. yl[t_] = InvereLaplaceTranform[L[],, t] 4 e-5 t - 5 e 4 t + 4 e 5 t - e e 5 t - 5 e 2+4 t HeaviideTheta[-2 + t] + e e 5 t - 5 e 4 (5+t) HeaviideTheta[-20 + t] - e 95 HeaviideTheta[-9 + t] - 4 e 5 t HeaviideTheta[-9 + t] + 5 e 9+4 t HeaviideTheta[-9 + t] + e 90 HeaviideTheta[-8 + t] + 4 e 5 t HeaviideTheta[-8 + t] - 5 e 8+4 t HeaviideTheta[-8 + t] - e 85 HeaviideTheta[-7 + t] - 4 e 5 t HeaviideTheta[-7 + t] + 5 e 7+4 t HeaviideTheta[-7 + t] + e 80 HeaviideTheta[-6 + t] + 4 e 5 t HeaviideTheta[-6 + t] - 5 e 4 (4+t) HeaviideTheta[-6 + t] - e 75 HeaviideTheta[-5 + t] - 4 e 5 t HeaviideTheta[-5 + t] + 5 e 5+4 t HeaviideTheta[-5 + t] + e 70 HeaviideTheta[-4 + t] + 4 e 5 t HeaviideTheta[-4 + t] - 5 e 4+4 t HeaviideTheta[-4 + t] - e 65 HeaviideTheta[-3 + t] - 4 e 5 t HeaviideTheta[-3 + t] + 5 e 3+4 t HeaviideTheta[-3 + t] + e 60 HeaviideTheta[-2 + t] + 4 e 5 t HeaviideTheta[-2 + t] - 5 e 4 (3+t) HeaviideTheta[-2 + t] - e 55 HeaviideTheta[- + t] - 4 e 5 t HeaviideTheta[- + t] + 5 e +4 t HeaviideTheta[- + t] + e 50 HeaviideTheta[-0 + t] + 4 e 5 t HeaviideTheta[-0 + t] - 5 e 0+4 t HeaviideTheta[-0 + t] - e 45 HeaviideTheta[-9 + t] - 4 e 5 t HeaviideTheta[-9 + t] + 5 e 9+4 t HeaviideTheta[-9 + t] + e 40 HeaviideTheta[-8 + t] + 4 e 5 t HeaviideTheta[-8 + t] - 5 e 8+4 t HeaviideTheta[-8 + t] - e 35 HeaviideTheta[-7 + t] - 4 e 5 t HeaviideTheta[-7 + t] + 5 e 7+4 t HeaviideTheta[-7 + t] + e 30 HeaviideTheta[-6 + t] + 4 e 5 t HeaviideTheta[-6 + t] - 5 e 6+4 t HeaviideTheta[-6 + t] - e 25 HeaviideTheta[-5 + t] - 4 e 5 t HeaviideTheta[-5 + t] + 5 e 5+4 t HeaviideTheta[-5 + t] + e 20 HeaviideTheta[-4 + t] + 4 e 5 t HeaviideTheta[-4 + t] - 5 e 4+4 t HeaviideTheta[-4 + t] - e 5 HeaviideTheta[-3 + t] - 4 e 5 t HeaviideTheta[-3 + t] + 5 e 3+4 t HeaviideTheta[-3 + t] + e 0 HeaviideTheta[-2 + t] + 4 e 5 t HeaviideTheta[-2 + t] - 5 e 2+4 t HeaviideTheta[-2 + t] - e 5 HeaviideTheta[- + t] - 4 e 5 t HeaviideTheta[- + t] + 5 e +4 t HeaviideTheta[- + t] Here i our olution plot. Notice the effect of turning on and turning off the driver force.

23 9-5-S-OverViewLaplaceTranform-StudentVerion.nb 23 psol = Plot[yl[t], {t, 0, 20}, PlotRange {{0, 20}, {0, 2}}] We examine the input function (quare wave) and the output function (rounded wave). Show[pRut, psol] Up cloe. Notice the delay of the ytem (ytem repone) in reponding to the turn on and to the hut off of the input function Show[pRut, psol, PlotRange {{0, 5.5}, {0, 6}}] There are many ue and reaon for uing Laplace Tranform, indeed, often

24 S-OverViewLaplaceTranform-StudentVerion.nb engineer live in the frequency domain of the tranform world, i.e., without returning to the time domain world of t, for they learn to "ee" and undertand in the frequency world and, a we aw, it i algebra and not calculu - thu making it more elementary.