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[MUG] Laplace Transforms

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[MUG] Laplace Transforms
Author: Chuck Baker    Posted: 23/11/2000 21:56:58 GMT
>> From: Chuck Baker Hello all, I have the following linear set of first order differential equations which I have solved by coupling Laplace transforms with linear algebra. y' = 2y - 4z, y(0) = 3 z' = y - 3z, z(0) = 0 I derived solutions by hand by first taking the Laplace transforms of the differential equations, then, I solved the simultaneous equations for Y(s) and Z(s). Finally, I determined the inverse transforms to develop the solution expressions y(t) and z(t). I computed the solutions to be: Y(t) = 4e^t - e^-2t Z(t) = e^t - e^-2t How can this process be written in Maple? Thanks, Chuck

[MUG] Re: Laplace Transforms
Author: Maple Group    Posted: 27/11/2000 22:11:15 GMT
>> From: Maple Group | >> From: Chuck Baker | I have the following linear set of first order differential equations which | I have solved by coupling Laplace transforms with linear algebra. | y' = 2y - 4z, y(0) = 3 | z' = y - 3z, z(0) = 0 | ... | How can this process be written in Maple? -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Fri, 24 Nov 2000 08:50:52 -0800 From: Allan Wittkopf To: Subject: Laplace Transforms You could do: > eqns:={diff(y(t),t)=2*y(t)-4*z(t), y(0)=3, > diff(z(t),t)=y(t)-3*z(t),z(0)=0}; eqns := d d {-- y(t) = 2 y(t) - 4 z(t), y(0) = 3, -- z(t) = y(t) - 3 z(t), z(0) = 0} dt dt > dsolve(eqns,{y(t),z(t)},method=laplace); bytes used=1070064, alloc=982860, time=0.23 {y(t) = -exp(-2 t) + 4 exp(t), z(t) = -exp(-2 t) + exp(t)} > Allan Wittkopf -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- From: "Edgardo S. Cheb-Terrab" To: Subject: Laplace Transforms Date: Fri, 24 Nov 2000 09:36:40 -0800 As follows, Your ODEs: > sys := diff(y(t),t)=2*y(t)-4*z(t), diff(z(t),t)=y(t)-3*z(t); sys := y' = 2 y - 4 z, z' = y - 3 z Your initial conditions: > ics := y(0) = 3, z(0) = 0; ics := y(0) = 3, z(0) = 0 Call directly the ODE (exact) solver: > sol := dsolve({sys,ics}); sol := {y = 4 exp(t) - exp(-2 t), z = exp(t) - exp(-2 t)} That's it. It is interesting to have in mind that you can test these solutions; e.g. as follows > odetest(sol,[sys]); # the solution cancels the ODE system [0, 0] > map(limit,sol,t=0); # and satisfy the given initial conditions {y(0) = 3, z(0) = 0} Hope this is of use, Edgardo ___________________________________________________________________________ Edgardo S. Cheb-Terrab http://lie.uwaterloo.ca/ecterrab Centre for Experimental and Constructive Mathematics SFU, Canada Theoretical Physics Department UERJ,Brazil ___________________________________________________________________________ -- "Whoever has the most fun wins" -- -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Mon, 27 Nov 2000 15:17:41 -0500 From: "Douglas B. Meade" To: Subject: Laplace Transforms Dear MUG and George, Here is my solution in Maple 6.01. In earlier versions is might be necessary to explicitly load the transform package. This can be done with the command: > with( transform ); immediately after the restart command. > restart; > > sys := { diff( y(t), t ) = 2*y(t) - 4*z(t), > diff( z(t), t ) = y(t) - 3*z(t) }; d d sys := {-- y(t) = 2 y(t) - 4 z(t), -- z(t) = y(t) - 3 z(t)} dt dt > ic := { y(0)=3, z(0)=0 }; ic := {y(0) = 3, z(0) = 0} > > soln := dsolve( sys union ic, {y(t),z(t)}, method=laplace ); soln := {y(t) = -exp(-2 t) + 4 exp(t), z(t) = -exp(-2 t) + exp(t)} Note that the solution is a set of equations for the two components. Several recent MUG postings have talked about working with sets. The following commands provide two illustrations of the basic techniques for working with Maple sets. The individual components of this solution can be plotted as follows. > plot( subs( soln, [y(t),z(t)] ), t=0..2 ); The two components of the solution can be obtained as Maple functions. > Y := unapply( subs( soln, y(t) ), t ); Y := t -> -exp(-2 t) + 4 exp(t) > Z := unapply( subs( soln, z(t) ), t ); Z := t -> -exp(-2 t) + exp(t) I hope this answers your question and is of use to you. Doug ----------------------------------------------------------------------- Douglas B. Meade Phone: (803) 777-6183 FAX: (803) 777-6527 Department of Mathematics URL: http://www.math.sc.edu/~meade/ USC, Columbia, SC 29208 E-mail:

[MUG] Re: Laplace Transforms
Author: Robert Israel    Posted: 30/11/2000 22:50:49 GMT
>> From: Robert Israel > des:= {diff(y(t),t) = 2*y(t)-4*z(t),diff(z(t),t)=y(t)-3*z(t)}; ics:= {y(0)=3,z(0)=0}; Then you can solve it in one step: > dsolve(des union ics, {y(t),z(t)}); Or if you want to follow your steps using the Laplace transform: > with(inttrans): laplace(des,t,s); subs(ics,%); solve(%,{laplace(y(t),t,s),laplace(z(t),t,s)}); invlaplace(%,s,t); {z(t) = exp(t) - exp(-2 t), y(t) = -exp(-2 t) + 4 exp(t)} Robert Israel Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 On Thu, 23 Nov 2000, Chuck Baker wrote: | I derived solutions by hand by first taking the Laplace transforms of | the differential equations, then, I solved the simultaneous equations | for Y(s) and Z(s). Finally, I determined the inverse transforms to | develop the solution expressions y(t) and z(t). | | How can this process be written in Maple?

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