harmonic-oscillators-modelling/Harmonic_Oscillators.nb at main · acrypticcode/harmonic-oscillators-modelling · GitHub

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
(* Content-type: application/vnd.wolfram.mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 13.2' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       158,          7]
NotebookDataLength[    128951,       2725]
NotebookOptionsPosition[    124415,       2652]
NotebookOutlinePosition[    124811,       2668]
CellTagsIndexPosition[    124768,       2665]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["Harmonic Oscillators", "Title",
 CellChangeTimes->{{3.921865316372686*^9,
  3.921865320168757*^9}},ExpressionUUID->"ea584bd3-a576-472c-bf3a-\
6a15c1bb77ef"],

Cell["Free Harmonic Oscillators", "Subtitle",
 CellChangeTimes->{{3.9218653991848497`*^9,
  3.9218654038623333`*^9}},ExpressionUUID->"625ece21-5f4a-4149-8022-\
2d4e75c1d1fd"],

Cell["\<\
The governing equation of free harmonic oscillation is given by \
\>", "Text",
 CellChangeTimes->{{3.9218655846693172`*^9,
  3.921865594975662*^9}},ExpressionUUID->"dc723395-e4c7-4299-9d56-\
c1e8865ebcd0"],

Cell[TextData[Cell[BoxData[
 FormBox[
  RowBox[{
   RowBox[{
    RowBox[{"m",
     FractionBox[
      RowBox[{
       SuperscriptBox["d", "2"], "x"}],
      RowBox[{"\[DifferentialD]",
       SuperscriptBox["t", "2"]}]]}], "+",
    RowBox[{
     SuperscriptBox["\[Omega]", "2"], "x"}]}], "=", "0"}], TraditionalForm]],
 FormatType->
  TraditionalForm,ExpressionUUID->"daa7a10e-af9b-4391-b414-6be580b99f26"]], \
"Text",
 CellChangeTimes->{{3.921865617471236*^9, 3.921865708200677*^9}, {
  3.921867337258635*^9,
  3.92186733725873*^9}},ExpressionUUID->"cbb5f60d-9ba9-41a6-919a-\
3004ddadf13c"],

Cell["We solve this ordinary differential equation ", "Text",
 CellChangeTimes->{{3.921865752180674*^9,
  3.9218657585686617`*^9}},ExpressionUUID->"04b87d27-220f-4bae-88b8-\
4f380216624f"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"solution", "=",
   RowBox[{"DSolve", "[",
    RowBox[{
     RowBox[{"{",
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"m", "*",
          RowBox[{
           RowBox[{"x", "''"}], "[", "t", "]"}]}], "+",
         RowBox[{
          RowBox[{"\[Omega]", "^", "2"}], "*",
          RowBox[{"x", "[", "t", "]"}]}]}], "==", "0"}], ",",
       RowBox[{
        RowBox[{"x", "[", "0", "]"}], "==", "x0"}], ",",
       RowBox[{
        RowBox[{
         RowBox[{"x", "'"}], "[", "0", "]"}], "==", "v0"}]}], "}"}], ",",
     RowBox[{"x", "[", "t", "]"}], ",", "t"}], "]"}]}], "\n"}]], "Input",
 CellChangeTimes->{{3.9218657631405363`*^9, 3.921865763143734*^9}, {
  3.9218672787127542`*^9, 3.921867304473789*^9}},
 CellLabel->"In[40]:=",ExpressionUUID->"54a92104-b684-4aef-8203-a2c69c2eced2"],

Cell[BoxData[
 RowBox[{"{",
  RowBox[{"{",
   RowBox[{
    RowBox[{"x", "[", "t", "]"}], "\[Rule]",
    FractionBox[
     RowBox[{
      RowBox[{"x0", " ", "\[Omega]", " ",
       RowBox[{"Cos", "[",
        FractionBox[
         RowBox[{"t", " ", "\[Omega]"}],
         SqrtBox["m"]], "]"}]}], "+",
      RowBox[{
       SqrtBox["m"], " ", "v0", " ",
       RowBox[{"Sin", "[",
        FractionBox[
         RowBox[{"t", " ", "\[Omega]"}],
         SqrtBox["m"]], "]"}]}]}], "\[Omega]"]}], "}"}], "}"}]], "Output",
 CellChangeTimes->{
  3.92186577193754*^9, {3.921867117257621*^9, 3.92186712749634*^9}, {
   3.921867282621616*^9, 3.9218673412926693`*^9}, 3.921867399385055*^9},
 CellLabel->"Out[40]=",ExpressionUUID->"24d2ccfa-b9a8-4770-ab47-b031d7d8c5b9"]
}, Open  ]],

Cell["And plot the solution with mass m=1 and initial conditions  ", "Text",
 CellChangeTimes->{{3.921865803112472*^9, 3.9218658139784*^9}, {
  3.921866368314575*^9, 3.921866368441032*^9}, {3.92186732945564*^9,
  3.921867331938208*^9}},ExpressionUUID->"dccb7781-d63d-49ce-9c1e-\
393c0ae334d8"],

Cell[TextData[{
 "\[Omega] = 2\[Pi],  ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{
     SubscriptBox["x", "0"], "=", "1"}], ",",
    RowBox[{
     SubscriptBox["v", "0"], "=", "0"}]}], TraditionalForm]],
  FormatType->TraditionalForm,ExpressionUUID->
  "4839c6cd-bff2-40fd-b96b-846fe5d2bdd3"]
}], "Text",
 CellChangeTimes->{{3.9218658433462353`*^9,
  3.9218659193721933`*^9}},ExpressionUUID->"49f753ad-10c5-47a2-ae76-\
37839e6c9712"],

Cell["over the time interval 0\[LessEqual]t\[LessEqual]5", "Text",
 CellChangeTimes->{{3.921867171628138*^9,
  3.921867210432601*^9}},ExpressionUUID->"ff67ce21-c0da-4721-8545-\
620a233bc68f"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Plot", "[",
  RowBox[{
   RowBox[{"Evaluate", "[",
    RowBox[{
     RowBox[{
      RowBox[{"x", "[", "t", "]"}], "/.", " ", "solution"}], "/.", " ",
     RowBox[{"{",
      RowBox[{
       RowBox[{"m", "->", "1"}], ",",
       RowBox[{"\[Omega]", "->",
        RowBox[{"2", "*", "Pi"}]}], ",",
       RowBox[{"x0", "->", "1"}], ",",
       RowBox[{"v0", "->", "0"}]}], "}"}]}], "]"}], ",",
   RowBox[{"{",
    RowBox[{"t", ",", "0", ",", "5"}], "}"}], ",",
   RowBox[{"PlotLabel", "->", "\"\<Free Harmonic Oscillation\>\""}]}],
  "]"}]], "Input",
 CellChangeTimes->{{3.921865927846663*^9, 3.9218659385138693`*^9}, {
  3.921867098134972*^9, 3.921867125184779*^9}, {3.921867308320323*^9,
  3.921867310824175*^9}},
 CellLabel->"In[41]:=",ExpressionUUID->"ae14a8a5-73d9-4e17-8b31-5f6da6148fab"],

Cell[BoxData[
 GraphicsBox[{{{}, {},
    TagBox[
     {RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6], Opacity[
      1.], LineBox[CompressedData["
1:eJwU23c8ld8fAHC59rxGSxKpRFZl5Et9TqVUlLJKqSQ7oSKSKKuISMooNCjK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       "]]},
     Annotation[#, "Charting`Private`Tag$41083#1"]& ]}, {}},
  AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
  Axes->{True, True},
  AxesLabel->{None, None},
  AxesOrigin->{0, 0},
  DisplayFunction->Identity,
  Frame->{{False, False}, {False, False}},
  FrameLabel->{{None, None}, {None, None}},
  FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
  GridLines->{None, None},
  GridLinesStyle->Directive[
    GrayLevel[0.5, 0.4]],
  ImagePadding->All,
  Method->{
   "DefaultBoundaryStyle" -> Automatic,
    "DefaultGraphicsInteraction" -> {
     "Version" -> 1.2, "TrackMousePosition" -> {True, False},
      "Effects" -> {
       "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2},
        "Droplines" -> {
         "freeformCursorMode" -> True,
          "placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" ->
    AbsolutePointSize[6], "ScalingFunctions" -> None,
    "CoordinatesToolOptions" -> {"DisplayFunction" -> ({
        (Identity[#]& )[
         Part[#, 1]],
        (Identity[#]& )[
         Part[#, 2]]}& ), "CopiedValueFunction" -> ({
        (Identity[#]& )[
         Part[#, 1]],
        (Identity[#]& )[
         Part[#, 2]]}& )}},
  PlotLabel->FormBox["\"Free Harmonic Oscillation\"", TraditionalForm],
  PlotRange->{{0, 5}, {-0.9999981778376758, 0.9999999999997945}},
  PlotRangeClipping->True,
  PlotRangePadding->{{
     Scaled[0.02],
     Scaled[0.02]}, {
     Scaled[0.05],
     Scaled[0.05]}},
  Ticks->{Automatic, Automatic}]], "Output",
 CellChangeTimes->{
  3.921865945826292*^9, {3.921867119367475*^9, 3.921867130603093*^9}, {
   3.921867293228568*^9, 3.9218673449642267`*^9}, 3.9218674013783693`*^9},
 CellLabel->"Out[41]=",ExpressionUUID->"28350e8e-ab4c-4dd2-9196-fbe032c711cd"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Damped Harmonic Oscillators ", "Subtitle",
 CellChangeTimes->{{3.9218659607950687`*^9,
  3.921865965137466*^9}},ExpressionUUID->"d7ac7a06-f01b-4699-b311-\
707b22c5c475"],

Cell["\<\
The governing equation of damped harmonic oscillation is given by \
\>", "Text",
 CellChangeTimes->{{3.9218660942309713`*^9,
  3.92186610293834*^9}},ExpressionUUID->"65c4b946-849d-4e98-b1c1-\
383d03ad7490"],

Cell[TextData[{
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{"m",
     FractionBox[
      RowBox[{
       SuperscriptBox["\[DifferentialD]", "2"], "x"}],
      RowBox[{"\[DifferentialD]",
       SuperscriptBox["t", "2"]}]]}], "+",
    RowBox[{"\[Beta]",
     FractionBox[
      RowBox[{"\[DifferentialD]", "x"}],
      RowBox[{"\[DifferentialD]", "t"}]]}]}], TraditionalForm]],
  FormatType->TraditionalForm,ExpressionUUID->
  "d8cdd690-003b-47a1-8d87-599d45b2c115"],
 "+",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{
     SuperscriptBox["\[Omega]", "2"], "x"}], "=", "0"}], TraditionalForm]],
  FormatType->TraditionalForm,ExpressionUUID->
  "78cd5400-70e3-483d-81e5-0c65e3e58b9e"]
}], "Text",
 CellChangeTimes->{{3.9218661483485928`*^9, 3.921866212099921*^9}, {
  3.921867347472013*^9,
  3.921867348852726*^9}},ExpressionUUID->"349263fa-d973-4173-bf19-\
6fc9412cf946"],

Cell["We solve this ordinary differential equation ", "Text",
 CellChangeTimes->{{3.9218662256086473`*^9,
  3.921866232862525*^9}},ExpressionUUID->"229e73f6-c713-480e-89c4-\
d9688d93e372"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"solution", "=",
  RowBox[{"DSolve", "[",
   RowBox[{
    RowBox[{"{",
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"m", "*",
         RowBox[{
          RowBox[{"x", "''"}], "[", "t", "]"}]}], "+",
        RowBox[{"\[Beta]", "*",
         RowBox[{
          RowBox[{"x", "'"}], "[", "t", "]"}]}], "+",
        RowBox[{
         RowBox[{"\[Omega]", "^", "2"}], "*",
         RowBox[{"x", "[", "t", "]"}]}]}], "==", "0"}], ",",
      RowBox[{
       RowBox[{"x", "[", "0", "]"}], "==", "x0"}], ",",
      RowBox[{
       RowBox[{
        RowBox[{"x", "'"}], "[", "0", "]"}], "==", "v0"}]}], "}"}], ",",
    RowBox[{"x", "[", "t", "]"}], ",", "t"}], "]"}]}]], "Input",
 CellChangeTimes->{{3.921866244613833*^9, 3.921866244616914*^9}, {
  3.9218663190779133`*^9, 3.921866321488759*^9}, {3.921867351183263*^9,
  3.92186735188594*^9}},
 CellLabel->"In[42]:=",ExpressionUUID->"64641f53-d3cc-4848-8b02-f38fd86d79f1"],

Cell[BoxData[
 RowBox[{"{",
  RowBox[{"{",
   RowBox[{
    RowBox[{"x", "[", "t", "]"}], "\[Rule]",
    RowBox[{
     FractionBox["1",
      RowBox[{"2", " ",
       SqrtBox[
        RowBox[{
         SuperscriptBox["\[Beta]", "2"], "-",
         RowBox[{"4", " ", "m", " ",
          SuperscriptBox["\[Omega]", "2"]}]}]]}]],
     RowBox[{"(",
      RowBox[{
       RowBox[{
        RowBox[{"-", "2"}], " ",
        SuperscriptBox["\[ExponentialE]",
         RowBox[{
          FractionBox["1", "2"], " ", "t", " ",
          RowBox[{"(",
           RowBox[{
            RowBox[{"-",
             FractionBox["\[Beta]", "m"]}], "-",
            FractionBox[
             SqrtBox[
              RowBox[{
               SuperscriptBox["\[Beta]", "2"], "-",
               RowBox[{"4", " ", "m", " ",
                SuperscriptBox["\[Omega]", "2"]}]}]], "m"]}], ")"}]}]], " ",
        "m", " ", "v0"}], "+",
       RowBox[{"2", " ",
        SuperscriptBox["\[ExponentialE]",
         RowBox[{
          FractionBox["1", "2"], " ", "t", " ",
          RowBox[{"(",
           RowBox[{
            RowBox[{"-",
             FractionBox["\[Beta]", "m"]}], "+",
            FractionBox[
             SqrtBox[
              RowBox[{
               SuperscriptBox["\[Beta]", "2"], "-",
               RowBox[{"4", " ", "m", " ",
                SuperscriptBox["\[Omega]", "2"]}]}]], "m"]}], ")"}]}]], " ",
        "m", " ", "v0"}], "-",
       RowBox[{
        SuperscriptBox["\[ExponentialE]",
         RowBox[{
          FractionBox["1", "2"], " ", "t", " ",
          RowBox[{"(",
           RowBox[{
            RowBox[{"-",
             FractionBox["\[Beta]", "m"]}], "-",
            FractionBox[
             SqrtBox[
              RowBox[{
               SuperscriptBox["\[Beta]", "2"], "-",
               RowBox[{"4", " ", "m", " ",
                SuperscriptBox["\[Omega]", "2"]}]}]], "m"]}], ")"}]}]], " ",
        "x0", " ", "\[Beta]"}], "+",
       RowBox[{
        SuperscriptBox["\[ExponentialE]",
         RowBox[{
          FractionBox["1", "2"], " ", "t", " ",
          RowBox[{"(",
           RowBox[{
            RowBox[{"-",
             FractionBox["\[Beta]", "m"]}], "+",
            FractionBox[
             SqrtBox[
              RowBox[{
               SuperscriptBox["\[Beta]", "2"], "-",
               RowBox[{"4", " ", "m", " ",
                SuperscriptBox["\[Omega]", "2"]}]}]], "m"]}], ")"}]}]], " ",
        "x0", " ", "\[Beta]"}], "+",
       RowBox[{
        SuperscriptBox["\[ExponentialE]",
         RowBox[{
          FractionBox["1", "2"], " ", "t", " ",
          RowBox[{"(",
           RowBox[{
            RowBox[{"-",
             FractionBox["\[Beta]", "m"]}], "-",
            FractionBox[
             SqrtBox[
              RowBox[{
               SuperscriptBox["\[Beta]", "2"], "-",
               RowBox[{"4", " ", "m", " ",
                SuperscriptBox["\[Omega]", "2"]}]}]], "m"]}], ")"}]}]], " ",
        "x0", " ",
        SqrtBox[
         RowBox[{
          SuperscriptBox["\[Beta]", "2"], "-",
          RowBox[{"4", " ", "m", " ",
           SuperscriptBox["\[Omega]", "2"]}]}]]}], "+",
       RowBox[{
        SuperscriptBox["\[ExponentialE]",
         RowBox[{
          FractionBox["1", "2"], " ", "t", " ",
          RowBox[{"(",
           RowBox[{
            RowBox[{"-",
             FractionBox["\[Beta]", "m"]}], "+",
            FractionBox[
             SqrtBox[
              RowBox[{
               SuperscriptBox["\[Beta]", "2"], "-",
               RowBox[{"4", " ", "m", " ",
                SuperscriptBox["\[Omega]", "2"]}]}]], "m"]}], ")"}]}]], " ",
        "x0", " ",
        SqrtBox[
         RowBox[{
          SuperscriptBox["\[Beta]", "2"], "-",
          RowBox[{"4", " ", "m", " ",
           SuperscriptBox["\[Omega]", "2"]}]}]]}]}], ")"}]}]}], "}"}],
  "}"}]], "Output",
 CellChangeTimes->{
  3.921866326537346*^9, 3.921867133344792*^9, {3.9218673523787127`*^9,
   3.921867366210519*^9}, 3.921867403479203*^9},
 CellLabel->"Out[42]=",ExpressionUUID->"8e40eb45-f786-4e43-883d-87e5f57f78a2"]
}, Open  ]],

Cell["\<\
And plot the solution with mass m=1 and  initial conditions  \
\>", "Text",
 CellChangeTimes->{{3.921866342792461*^9, 3.9218663610479097`*^9}, {
  3.921867356667161*^9,
  3.9218673600933228`*^9}},ExpressionUUID->"17a8bae8-7bb0-471b-9500-\
bf89a6fec6e5"],

Cell[TextData[{
 "\[Omega] = 2\[Pi],  \[Beta]=1, ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{
     SubscriptBox["x", "0"], "=", "1"}], ",",
    RowBox[{
     SubscriptBox["v", "0"], "=", "0"}]}], TraditionalForm]],
  FormatType->TraditionalForm,ExpressionUUID->
  "38dad8ae-013a-4a5f-bca7-e33d89dca27b"]
}], "Text",
 CellChangeTimes->{{3.921866385838161*^9,
  3.9218664001167707`*^9}},ExpressionUUID->"887ee468-05a7-4e95-a956-\
560ab6b88361"],

Cell["over the time interval 0\[LessEqual]t\[LessEqual]5", "Text",
 CellChangeTimes->{
  3.92186722452831*^9},ExpressionUUID->"656c51c9-ccee-4d6f-a06c-5017c5e5ba94"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Plot", "[",
  RowBox[{
   RowBox[{"Evaluate", "[",
    RowBox[{
     RowBox[{
      RowBox[{"x", "[", "t", "]"}], "/.", " ", "solution"}], "/.", " ",
     RowBox[{"{",
      RowBox[{
       RowBox[{"m", "->", "1"}], ",",
       RowBox[{"\[Omega]", "->",
        RowBox[{"2", "*", "Pi"}]}], ",",
       RowBox[{"\[Beta]", "->", "1"}], ",",
       RowBox[{"x0", "->", "1"}], ",",
       RowBox[{"v0", "->", "0"}]}], "}"}]}], "]"}], ",",
   RowBox[{"{",
    RowBox[{"t", ",", "0", ",", "5"}], "}"}], ",",
   RowBox[{"PlotLabel", "->", "\"\<Damped Harmonic Oscillation\>\""}]}],
  "]"}]], "Input",
 CellChangeTimes->{{3.921866404664896*^9, 3.921866424219096*^9}, {
  3.921867136343957*^9, 3.921867136565156*^9}, {3.9218673689341583`*^9,
  3.921867371236514*^9}},
 CellLabel->"In[36]:=",ExpressionUUID->"bc6464ad-4011-4185-9ac2-d8a63cc24dab"],

Cell[BoxData[
 GraphicsBox[{{{}, {},
    TagBox[
     {RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6], Opacity[
      1.], LineBox[CompressedData["
1:eJwUV3c81W8bxjEODo7RklDKjrK17kcIpahQREqyMyoNVKJSUuRnV6hIRGVH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