(* Content-type: application/vnd.wolfram.mathematica *)

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

(* CreatedBy='Mathematica 8.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       157,          7]
NotebookDataLength[     15446,        475]
NotebookOptionsPosition[     14034,        422]
NotebookOutlinePosition[     14418,        438]
CellTagsIndexPosition[     14375,        435]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["Az extravaszkul\[AAcute]ris gy\[OAcute]gyszeradagol\[AAcute]s modellj\
\[EAcute]nek vizsg\[AAcute]lata", "MainSection",
 CellChangeTimes->{{3.4948424149375*^9, 3.49484245203125*^9}}],

Cell[CellGroupData[{

Cell["Modell", "Section",
 CellChangeTimes->{{3.494843613859375*^9, 3.49484361471875*^9}}],

Cell["\<\
Tudjuk (illetve tanulni fogjuk), hogy extravaszkul\[AAcute]ris \
adagol\[AAcute]s eset\[EAcute]n a v\[EAcute]rben a gy\[OAcute]gyszer koncentr\
\[AAcute]ci\[OAcute]t az al\[AAcute]bbi modell \[IAcute]rja le: \
\>", "Text",
 CellChangeTimes->{{3.494842454296875*^9, 3.4948425533125*^9}}],

Cell[TextData[{
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{"x", "'"}], " ", "=", " ", 
    RowBox[{
     RowBox[{"-", 
      SubscriptBox["k", "1"]}], " ", "x"}]}], TraditionalForm]]],
 ", ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{"y", "'"}], "=", 
    RowBox[{
     RowBox[{
      SubscriptBox["k", "1"], " ", "x"}], "-", 
     RowBox[{
      SubscriptBox["k", "2"], "y"}]}]}], TraditionalForm]]]
}], "Text",
 CellChangeTimes->{{3.49484256590625*^9, 3.4948426431875*^9}}],

Cell[TextData[{
 "ahol ",
 Cell[BoxData[
  FormBox[
   RowBox[{"x", "(", "t", ")"}], TraditionalForm]]],
 " a gy\[OAcute]gyszer koncentr\[AAcute]ci\[OAcute]ja a gyomorban,",
 Cell[BoxData[
  FormBox[
   RowBox[{" ", 
    RowBox[{"y", "(", "t", ")"}]}], TraditionalForm]]],
 " pedig a v\[EAcute]rben, ahol  ",
 Cell[BoxData[
  FormBox[
   SubscriptBox["k", "1"], TraditionalForm]]],
 "  a felsz\[IAcute]v\[OAcute]d\[AAcute]si, ",
 Cell[BoxData[
  FormBox[
   SubscriptBox["k", "2"], TraditionalForm]]],
 " az elimin\[AAcute]ci\[OAcute]s r\[AAcute]ta."
}], "Text",
 CellChangeTimes->{{3.49484264659375*^9, 3.49484280978125*^9}, {
  3.494842891875*^9, 3.494842894453125*^9}, {3.4948429543125*^9, 
  3.494842984703125*^9}, {3.49484363034375*^9, 3.4948436325*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Megold\[AAcute]s", "Section",
 CellChangeTimes->{{3.4948436350625*^9, 3.494843637328125*^9}}],

Cell[TextData[{
 "Ha a ",
 Cell[BoxData[
  FormBox[
   RowBox[{"t", "=", "0"}], TraditionalForm]]],
 "  pillanatban ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{
     RowBox[{"x", "(", "0", ")"}], "=", 
     RowBox[{
      SubscriptBox["x", "0"], ">", "0"}]}], ",", " ", 
    RowBox[{
     RowBox[{"\[EAcute]s", " ", 
      RowBox[{"y", "(", "0", ")"}]}], "=", "0", " "}]}], TraditionalForm]]],
 " a kezdeti \[EAcute]rt\[EAcute]k, akkor a megold\[AAcute]s explicit alakban:"
}], "Text",
 CellChangeTimes->{{3.49484264659375*^9, 3.49484280978125*^9}, {
   3.494842891875*^9, 3.494842894453125*^9}, {3.4948429543125*^9, 
   3.494842984703125*^9}, 3.49484363034375*^9}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Simplify", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"x", "[", "t", "]"}], ",", 
     RowBox[{"y", "[", "t", "]"}]}], "}"}], "/.", 
   RowBox[{"DSolve", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"x", "'"}], "[", "t", "]"}], "\[Equal]", 
        RowBox[{
         RowBox[{"-", "k1"}], " ", 
         RowBox[{"x", "[", "t", "]"}]}]}], ",", 
       RowBox[{
        RowBox[{
         RowBox[{"y", "'"}], "[", "t", "]"}], "\[Equal]", 
        RowBox[{
         RowBox[{"k1", " ", 
          RowBox[{"x", "[", "t", "]"}]}], "-", 
         RowBox[{"k2", " ", 
          RowBox[{"y", "[", "t", "]"}]}]}]}], ",", 
       RowBox[{
        RowBox[{"x", "[", "0", "]"}], "\[Equal]", "x0"}], ",", 
       RowBox[{
        RowBox[{"y", "[", "0", "]"}], "\[Equal]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"x", "[", "t", "]"}], ",", 
       RowBox[{"y", "[", "t", "]"}]}], "}"}], ",", "t"}], "]"}]}], 
  "]"}]], "Input",
 CellChangeTimes->{{3.49484281303125*^9, 3.494842922703125*^9}}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{
     SuperscriptBox["\[ExponentialE]", 
      RowBox[{
       RowBox[{"-", "k1"}], " ", "t"}]], " ", "x0"}], ",", 
    RowBox[{"-", 
     FractionBox[
      RowBox[{
       RowBox[{"(", 
        RowBox[{
         SuperscriptBox["\[ExponentialE]", 
          RowBox[{
           RowBox[{"-", "k1"}], " ", "t"}]], "-", 
         SuperscriptBox["\[ExponentialE]", 
          RowBox[{
           RowBox[{"-", "k2"}], " ", "t"}]]}], ")"}], " ", "k1", " ", "x0"}], 
      RowBox[{"k1", "-", "k2"}]]}]}], "}"}], "}"}]], "Output",
 CellChangeTimes->{
  3.494842881515625*^9, {3.49484291321875*^9, 3.49484292371875*^9}}]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Grafikus vizsg\[AAcute]lat", "Section",
 CellChangeTimes->{{3.49484364571875*^9, 3.494843649671875*^9}}],

Cell[TextData[{
 "Vizsg\[AAcute]ljuk meg az ",
 Cell[BoxData[
  FormBox[
   RowBox[{"y", "(", "t", ")"}], TraditionalForm]]],
 " v\[EAcute]rszintg\[ODoubleDot]rbe alakul\[AAcute]s\[AAcute]t a param\
\[EAcute]terek f\[UDoubleDot]gg\[EAcute]ny\[EAcute]ben:"
}], "Text",
 CellChangeTimes->{{3.494842987578125*^9, 3.494843056*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"v\[EAcute]rszin", "[", "t_", "]"}], ":=", 
  RowBox[{"-", 
   FractionBox[
    RowBox[{
     RowBox[{"(", 
      RowBox[{
       SuperscriptBox["\[ExponentialE]", 
        RowBox[{
         RowBox[{"-", "k1"}], " ", "t"}]], "-", 
       SuperscriptBox["\[ExponentialE]", 
        RowBox[{
         RowBox[{"-", "k2"}], " ", "t"}]]}], ")"}], " ", "k1", " ", "x0"}], 
    RowBox[{"k1", "-", "k2"}]]}]}]], "Input",
 CellChangeTimes->{{3.494843059890625*^9, 3.494843077921875*^9}, 
   3.494843459890625*^9}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Manipulate", "[", 
  RowBox[{
   RowBox[{"Quiet", "[", 
    RowBox[{"Plot", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
        RowBox[{
         SuperscriptBox["\[ExponentialE]", 
          RowBox[{
           RowBox[{"-", "k1"}], " ", "t"}]], " ", "x0"}], ",", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{
           RowBox[{"(", 
            RowBox[{
             SuperscriptBox["\[ExponentialE]", 
              RowBox[{
               RowBox[{"-", "k1"}], " ", "t"}]], "-", 
             SuperscriptBox["\[ExponentialE]", 
              RowBox[{
               RowBox[{"-", "k2"}], " ", "t"}]]}], ")"}], " ", "k1", " ", 
           "x0"}], 
          RowBox[{"k1", "-", "k2"}]]}]}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{"t", ",", "0", ",", "5"}], "}"}], ",", 
      RowBox[{"AxesOrigin", "->", 
       RowBox[{"{", 
        RowBox[{"0", ",", "0"}], "}"}]}], ",", 
      RowBox[{"PlotRange", "\[Rule]", 
       RowBox[{"{", 
        RowBox[{"0", ",", "x0"}], "}"}]}]}], "]"}], "]"}], ",", 
   "\[IndentingNewLine]", 
   RowBox[{"{", 
    RowBox[{"x0", ",", "0", ",", "1"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"k1", ",", "0", ",", "4"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"k2", ",", "0", ",", "k1"}], "}"}], ",", 
   RowBox[{"ControlPlacement", "\[Rule]", "Left"}], ",", 
   RowBox[{"SaveDefinitions", "\[Rule]", "True"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.494843095703125*^9, 3.4948432145625*^9}, {
  3.4948432550625*^9, 3.494843268984375*^9}, {3.494843335421875*^9, 
  3.494843376953125*^9}, {3.494843408375*^9, 3.494843422171875*^9}, {
  3.49484354834375*^9, 3.494843560828125*^9}}],

Cell[BoxData[
 TagBox[
  StyleBox[
   DynamicModuleBox[{$CellContext`k1$$ = 0.85, $CellContext`k2$$ = 
    0.15214999999999998`, $CellContext`x0$$ = 0.212, Typeset`show$$ = True, 
    Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", 
    Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = 
    "\"untitled\"", Typeset`specs$$ = {{
      Hold[$CellContext`x0$$], 0, 1}, {
      Hold[$CellContext`k1$$], 0, 4}, {
      Hold[$CellContext`k2$$], 0, 
      Dynamic[$CellContext`k1$$]}}, Typeset`size$$ = {376., {110., 128.}}, 
    Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = 
    True, $CellContext`x0$140579$$ = 0, $CellContext`k1$140580$$ = 
    0, $CellContext`k2$140581$$ = 0}, 
    DynamicBox[Manipulate`ManipulateBoxes[
     1, StandardForm, 
      "Variables" :> {$CellContext`k1$$ = 0, $CellContext`k2$$ = 
        0, $CellContext`x0$$ = 0}, "ControllerVariables" :> {
        Hold[$CellContext`x0$$, $CellContext`x0$140579$$, 0], 
        Hold[$CellContext`k1$$, $CellContext`k1$140580$$, 0], 
        Hold[$CellContext`k2$$, $CellContext`k2$140581$$, 0]}, 
      "OtherVariables" :> {
       Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, 
        Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, 
        Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$,
         Typeset`skipInitDone$$}, "Body" :> Quiet[
        Plot[{
         E^((-$CellContext`k1$$) $CellContext`t) $CellContext`x0$$, -(((
             E^((-$CellContext`k1$$) $CellContext`t) - 
             E^((-$CellContext`k2$$) $CellContext`t)) $CellContext`k1$$) \
($CellContext`x0$$/($CellContext`k1$$ - $CellContext`k2$$)))}, \
{$CellContext`t, 0, 5}, AxesOrigin -> {0, 0}, 
         PlotRange -> {0, $CellContext`x0$$}]], 
      "Specifications" :> {{$CellContext`x0$$, 0, 1}, {$CellContext`k1$$, 0, 
         4}, {$CellContext`k2$$, 0, 
         Dynamic[$CellContext`k1$$]}}, 
      "Options" :> {ControlPlacement -> Left}, "DefaultOptions" :> {}],
     ImageSizeCache->{885., {275., 284.}},
     SingleEvaluation->True],
    Deinitialization:>None,
    DynamicModuleValues:>{},
    SynchronousInitialization->True,
    UnsavedVariables:>{Typeset`initDone$$},
    UntrackedVariables:>{Typeset`size$$}], "Manipulate",
   Deployed->True,
   StripOnInput->False],
  Manipulate`InterpretManipulate[1]]], "Output",
 CellChangeTimes->{
  3.494843172125*^9, 3.494843215125*^9, {3.49484326396875*^9, 
   3.494843270171875*^9}, {3.4948433489375*^9, 3.494843377515625*^9}, {
   3.494843418203125*^9, 3.494843422734375*^9}, 3.494843562328125*^9}]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Hol van a v\[EAcute]rszintg\[ODoubleDot]rb\[EAcute]nek sz\[EAcute]ls\
\[ODoubleAcute]\[EAcute]rt\[EAcute]ke?", "Section",
 CellChangeTimes->{{3.494843467890625*^9, 3.494843478328125*^9}, 
   3.52424202083924*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"verszint", "[", "t_", "]"}], ":=", 
  RowBox[{"-", 
   FractionBox[
    RowBox[{
     RowBox[{"(", 
      RowBox[{
       SuperscriptBox["\[ExponentialE]", 
        RowBox[{
         RowBox[{"-", "k1"}], " ", "t"}]], "-", 
       SuperscriptBox["\[ExponentialE]", 
        RowBox[{
         RowBox[{"-", "k2"}], " ", "t"}]]}], ")"}], " ", "k1", " ", "x0"}], 
    RowBox[{"k1", "-", "k2"}]]}]}]], "Input",
 CellChangeTimes->{{3.494843059890625*^9, 3.494843077921875*^9}, 
   3.494843459890625*^9, {3.4948435046875*^9, 3.494843505265625*^9}, 
   3.494843540609375*^9}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"verszint", "'"}], "[", "t", "]"}]], "Input",
 CellChangeTimes->{{3.494843532265625*^9, 3.494843541859375*^9}}],

Cell[BoxData[
 RowBox[{"-", 
  FractionBox[
   RowBox[{"k1", " ", 
    RowBox[{"(", 
     RowBox[{
      RowBox[{
       RowBox[{"-", 
        SuperscriptBox["\[ExponentialE]", 
         RowBox[{
          RowBox[{"-", "k1"}], " ", "t"}]]}], " ", "k1"}], "+", 
      RowBox[{
       SuperscriptBox["\[ExponentialE]", 
        RowBox[{
         RowBox[{"-", "k2"}], " ", "t"}]], " ", "k2"}]}], ")"}], " ", "x0"}], 
   RowBox[{"k1", "-", "k2"}]]}]], "Output",
 CellChangeTimes->{{3.4948435331875*^9, 3.49484354409375*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Solve", "[", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"verszint", "'"}], "[", "t", "]"}], "\[Equal]", "0"}], ",", 
   "t"}], "]"}]], "Input",
 CellChangeTimes->{{3.49484349571875*^9, 3.49484353696875*^9}, {
  3.494843569328125*^9, 3.494843589984375*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Solve", "::", "\<\"ifun\"\>"}], ":", 
  " ", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Inverse functions are being used by \
\\\\\\\"\\\", \\\"MT\\\"]\\)\[NoBreak]\\!\\(\\*StyleBox[\\!\\(Solve\\), \
\\\"MT\\\"]\\)\[NoBreak]\\!\\(\\*StyleBox[\\\"\\\\\\\", so some solutions may \
not be found; use Reduce for complete solution information.\\\\\\\"\\\", \
\\\"MT\\\"]\\) \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", \
ButtonStyle->\\\"Link\\\", ButtonFrame->None, \
ButtonData:>\\\"paclet:ref/message/Solve/ifun\\\", ButtonNote -> \
\\\"Solve::ifun\\\"]\\)\"\>"}]], "Message", "MSG",
 CellChangeTimes->{3.494843590578125*^9}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"{", 
   RowBox[{"t", "\[Rule]", 
    FractionBox[
     RowBox[{"Log", "[", 
      FractionBox["k1", "k2"], "]"}], 
     RowBox[{"k1", "-", "k2"}]]}], "}"}], "}"}]], "Output",
 CellChangeTimes->{3.494843590578125*^9}]
}, Open  ]]
}, Open  ]],

Cell[TextData[{
 ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]",
  BaseStyle->"SlidePreviousNextLink",
  Appearance->{Automatic, None},
  ButtonFunction:>FrontEndExecute[{
     FrontEndToken[
      FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}],
  ButtonNote->FEPrivate`FrontEndResource[
   "FEStrings", "SlideshowPrevSlideText"]],
 "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]",
 ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]",
  BaseStyle->"SlidePreviousNextLink",
  Appearance->{Automatic, None},
  ButtonFunction:>FrontEndExecute[{
     FrontEndToken[
      FrontEnd`ButtonNotebook[], "ScrollPageNext"]}],
  ButtonNote->FEPrivate`FrontEndResource[
   "FEStrings", "SlideshowNextSlideText"]]
}], "Section",
 TextAlignment->Right]
}, Open  ]]
},
WindowSize->{1025, 706},
WindowMargins->{{127, Automatic}, {Automatic, 17}},
Magnification:>2. Inherited,
FrontEndVersion->"10.3 for Microsoft Windows (64-bit) (December 10, 2015)",
StyleDefinitions->"Modelbookstyle.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[579, 22, 188, 2, 126, "MainSection"],
Cell[CellGroupData[{
Cell[792, 28, 90, 1, 77, "Section"],
Cell[885, 31, 296, 5, 88, "Text"],
Cell[1184, 38, 490, 19, 55, "Text"],
Cell[1677, 59, 759, 22, 88, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[2473, 86, 99, 1, 77, "Section"],
Cell[2575, 89, 674, 20, 88, "Text"],
Cell[CellGroupData[{
Cell[3274, 113, 1105, 34, 132, "Input"],
Cell[4382, 149, 681, 21, 83, "Output"]
}, Open  ]]
}, Open  ]],
Cell[CellGroupData[{
Cell[5112, 176, 110, 1, 77, "Section"],
Cell[5225, 179, 327, 8, 55, "Text"],
Cell[5555, 189, 538, 16, 106, "Input"],
Cell[CellGroupData[{
Cell[6118, 209, 1676, 45, 235, "Input"],
Cell[7797, 256, 2594, 51, 589, "Output"]
}, Open  ]]
}, Open  ]],
Cell[CellGroupData[{
Cell[10440, 313, 218, 3, 77, "Section"],
Cell[10661, 318, 601, 17, 106, "Input"],
Cell[CellGroupData[{
Cell[11287, 339, 145, 3, 67, "Input"],
Cell[11435, 344, 520, 16, 83, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[11992, 365, 287, 8, 67, "Input"],
Cell[12282, 375, 651, 11, 123, "Message"],
Cell[12936, 388, 256, 8, 89, "Output"]
}, Open  ]]
}, Open  ]],
Cell[13219, 400, 799, 19, 77, "Section"]
}, Open  ]]
}
]
*)

(* End of internal cache information *)
