(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 260130, 4712] NotebookOptionsPosition[ 256132, 4596] NotebookOutlinePosition[ 256596, 4614] CellTagsIndexPosition[ 256553, 4611] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["06 Nov 09", "Section", CellChangeTimes->{{3.4349336471140003`*^9, 3.434933661432*^9}, 3.4663802709397*^9, {3.4663803337047*^9, 3.4663803347487*^9}}, FontSize->24], Cell["\<\ Here are the marching orders for the day. There are a couple of plots to \ print out and hand in with the homework. I recommend that you create another blank notebook, in which you'll create \ the cells you want to keep and print. You can always cut and paste \ expressions from here. Or just evaluate them here and cut and paste \ afterward.\ \>", "Text", CellChangeTimes->{{3.43496388391*^9, 3.4349639062019997`*^9}, { 3.434965187001*^9, 3.434965219019*^9}, {3.4664226016042*^9, 3.4664226653926*^9}, {3.4664232412936*^9, 3.4664232434152*^9}}, FontFamily->"Times New Roman", FontSize->24], Cell[CellGroupData[{ Cell["\<\ remember DSolve[] from last time ...\ \>", "Subsubsection", CellChangeTimes->{{3.434933693046*^9, 3.434933698547*^9}, {3.434933795977*^9, 3.434933796435*^9}, {3.434933876531*^9, 3.434933878119*^9}, 3.4349639155480003`*^9, 3.4663803036547003`*^9, {3.4664226737542*^9, 3.4664227009138002`*^9}, {3.4664227493050003`*^9, 3.4664227516762*^9}}, FontSize->24], Cell["\<\ One of the skills we worked on last week was the use of DSolve[]. Try it out \ on a couple of differential equations which appear on this week's homework, \ starting with the basic rocket equation:\ \>", "Text", CellChangeTimes->{{3.434933891527*^9, 3.434933934028*^9}, 3.434963836067*^9, {3.43496409392*^9, 3.434964100881*^9}, 3.435101548209*^9, 3.4663803460917*^9, {3.4664227445158*^9, 3.4664228062918*^9}, {3.4664228856958*^9, 3.4664228933710003`*^9}}, FontFamily->"Times New Roman", FontSize->24], Cell[BoxData[ RowBox[{"de1", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"v", "'"}], "[", "t", "]"}], " ", "\[Equal]", " ", RowBox[{ RowBox[{"-", "b"}], " ", RowBox[{"u", " ", "/", " ", RowBox[{"(", RowBox[{"m0", " ", "-", " ", RowBox[{"b", " ", "t"}]}], ")"}]}]}]}]}]], "Input", CellChangeTimes->{{3.4664228625298*^9, 3.466422878083*^9}}, FontSize->14], Cell[BoxData[ RowBox[{"bc1", "=", " ", RowBox[{ RowBox[{"v", "[", "0", "]"}], " ", "\[Equal]", "0"}]}]], "Input", CellChangeTimes->{{3.4664228991742*^9, 3.4664229062098*^9}, { 3.466422961465*^9, 3.4664229679858*^9}}], Cell["Your job: DSolve these for v[t].", "Text", CellChangeTimes->{{3.4664229098602*^9, 3.4664229351946*^9}}, FontSize->14], Cell["\<\ Similarly, for the boxcar problem you'll want to solve these:\ \>", "Text", CellChangeTimes->{{3.4664229381586*^9, 3.466422949843*^9}}, FontSize->14], Cell[BoxData[ RowBox[{"de2", " ", "=", " ", RowBox[{"F", "\[Equal]", " ", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"m0", " ", "+", " ", RowBox[{"b", " ", "t"}]}], ")"}], " ", RowBox[{ RowBox[{"v", "'"}], "[", "t", "]"}]}], " ", "+", " ", RowBox[{"b", " ", RowBox[{"v", "[", "t", "]"}]}]}]}]}]], "Input", CellChangeTimes->{{3.4664229509038*^9, 3.4664229789058*^9}, { 3.4664231525452003`*^9, 3.4664231720452003`*^9}}], Cell[BoxData[ RowBox[{"bc2", " ", "=", " ", RowBox[{ RowBox[{"v", "[", "0", "]"}], "\[Equal]", "0"}]}]], "Input", CellChangeTimes->{{3.4664231756956*^9, 3.4664231778952*^9}}], Cell["\<\ Copy the solutions to your notebook for printing later---that way you'll have \ it to consult when doing the homework, where you'll explain where this \ differential equation came from.\ \>", "Text", CellChangeTimes->{{3.466423220842*^9, 3.4664233084672003`*^9}}, FontSize->14] }, Open ]], Cell[CellGroupData[{ Cell["BTM problems", "Subsubsection", CellChangeTimes->{{3.434933693046*^9, 3.434933698547*^9}, {3.434933795977*^9, 3.434933796435*^9}, {3.434933876531*^9, 3.434933878119*^9}, 3.4349639155480003`*^9, 3.4663803036547003`*^9, 3.4664226737542*^9}, FontSize->24], Cell["Consider the function defined in Shankar:", "Text", CellChangeTimes->{{3.434933891527*^9, 3.434933934028*^9}, 3.434963836067*^9, {3.43496409392*^9, 3.434964100881*^9}, 3.435101548209*^9, 3.4663803460917*^9}, FontFamily->"Times New Roman", FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", RowBox[{"x_", ",", "y_"}], "]"}], " ", "=", " ", RowBox[{ RowBox[{"x", "^", "2"}], " ", "-", " ", RowBox[{"x", " ", "y"}], " ", "+", " ", RowBox[{"y", "^", "2"}]}]}]], "Input", CellChangeTimes->{{3.434933900592*^9, 3.434933924035*^9}}, FontSize->24], Cell["\<\ To visualize, try plotting the surface in 3 D. Notice you can rotate the figure with your mouse (hold down the left mouse \ button as you do so)\ \>", "Text", CellChangeTimes->{{3.434934311637*^9, 3.4349343329630003`*^9}, { 3.434964107051*^9, 3.43496416046*^9}, {3.4349645759560003`*^9, 3.4349645765629997`*^9}}, FontSize->24], Cell[BoxData[ RowBox[{"p1", "=", RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"x", ",", "y"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "6"}], ",", "6"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "6"}], ",", "6"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.434934382724*^9, 3.434934400461*^9}, {3.4349345643*^9, 3.434934564855*^9}, {3.434964179781*^9, 3.434964179906*^9}}, FontSize->24], Cell["Or to get the topographical map representation :", "Text", CellChangeTimes->{{3.4349344189230003`*^9, 3.4349344282790003`*^9}}, FontSize->24], Cell[BoxData[ RowBox[{"p2", "=", RowBox[{"ContourPlot", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"x", ",", "y"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "6"}], ",", "6"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "6"}], ",", "6"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.4349339648389997`*^9, 3.4349339884370003`*^9}, { 3.434934018499*^9, 3.434934018918*^9}, {3.43493428388*^9, 3.4349342912650003`*^9}, 3.43496420348*^9, {3.4349648759440002`*^9, 3.4349648761140003`*^9}}, FontSize->24], Cell["\<\ Your first assignment : add more contours to the plot. (?? ContourPlot will \ give you all the options.)\ \>", "Text", CellChangeTimes->{{3.4349340319519997`*^9, 3.4349340428380003`*^9}, { 3.4349340892279997`*^9, 3.4349340913*^9}, {3.4349341594519997`*^9, 3.4349341962320004`*^9}, {3.4349344373459997`*^9, 3.434934442019*^9}, { 3.434964215001*^9, 3.434964223043*^9}, {3.4349642715109997`*^9, 3.434964333226*^9}}, FontSize->24], Cell["Next, draw a circle of radius 5 centered at the origin:", "Text", CellChangeTimes->{{3.4349643470299997`*^9, 3.434964360962*^9}, 3.434964434316*^9, {3.4349652506099997`*^9, 3.434965260324*^9}}, FontSize->24], Cell[BoxData[{ RowBox[{ RowBox[{"x", "[", "t_", "]"}], " ", "=", " ", RowBox[{"5", " ", RowBox[{"Cos", "[", RowBox[{"2", " ", "\[Pi]", " ", "t"}], "]"}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"y", "[", "t_", "]"}], " ", "=", " ", RowBox[{"5", " ", RowBox[{"Sin", "[", RowBox[{"2", " ", "\[Pi]", " ", "t"}], "]"}]}]}]}], "Input", CellChangeTimes->{{3.434964487119*^9, 3.4349644913599997`*^9}, { 3.4349646178459997`*^9, 3.4349646516540003`*^9}}, FontSize->24], Cell[BoxData[ RowBox[{"p3", " ", "=", " ", RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"x", "[", "t", "]"}], ",", RowBox[{"y", "[", "t", "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "1"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.434934026817*^9, 3.434934026975*^9}, {3.434934199658*^9, 3.434934199965*^9}, {3.434934251757*^9, 3.434934257818*^9}, { 3.4349344435439997`*^9, 3.434934499736*^9}, {3.4349644441549997`*^9, 3.434964470526*^9}, {3.434964658243*^9, 3.434964684527*^9}, { 3.434964884251*^9, 3.434964884382*^9}}, FontSize->24], Cell["\<\ Your next task: make the line thicker and turn it red. (Check out the \ PlotStyle option.) And then show both of your plots together with:\ \>", "Text", CellChangeTimes->{{3.434964708189*^9, 3.434964766078*^9}}, FontFamily->"Times New Roman", FontSize->24], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"p2", ",", "p3"}], "]"}]], "Input", CellChangeTimes->{{3.434934512925*^9, 3.434934515617*^9}, {3.434964887455*^9, 3.434964893479*^9}}, FontSize->24], Cell["\<\ Next task: add a 3D curve (hmmm, maybe there is a 3D version of \ ParametricPlot) the Plot3D to get a figure like this one:\ \>", "Text", CellChangeTimes->{{3.434964804351*^9, 3.434964865559*^9}, {3.434964969152*^9, 3.434964980222*^9}, {3.434965021263*^9, 3.434965026175*^9}}, FontSize->24], Cell[BoxData[ Graphics3DBox[{GraphicsComplex3DBox[CompressedData[" 1:eJx1nXmcVcXR/u+ozCjKMsguAsIgimyyL8JpxGVYBASCikqMEVyJBo37EiGv vm4kEMQlrok/CJGoEH8kkZh7UATjAkZBiUQFRSeKAip6jUt49dznWyddIn9w P/e5Z/r0ebq6uqq6qs9Bp58/fuoehUJhU1WhsOfXn/+YevmqXbtapHx2u+OP i3btGhZGn3DUx7Vzmxr+3EOj6828LQnF2obt59ZUG97yhE1XdF94RDjjokfu 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One method: change variables to make it a one dimensional problem:\ \>", "Text", CellChangeTimes->{{3.434965042466*^9, 3.43496509495*^9}}, FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{"h", "[", "t_", "]"}], " ", "=", " ", RowBox[{"f", "[", RowBox[{ RowBox[{"x", "[", "t", "]"}], ",", RowBox[{"y", "[", "t", "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.434965096283*^9, 3.4349651032720003`*^9}}, FontSize->24], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"h", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.434965118514*^9, 3.4349651234630003`*^9}}, FontSize->24], Cell["\<\ Your assignment: find the extrema of h[t], and then armed with those special \ values of t, find the extremal points x[t],y[t]\ \>", "Text", CellChangeTimes->{{3.434965349013*^9, 3.434965388733*^9}, {3.434965511079*^9, 3.43496553462*^9}, {3.434965564755*^9, 3.434965579862*^9}}, FontSize->24], Cell["\<\ The Lagrange multiplier method ends up at the same place by defining the \ functions\ \>", "Text", CellChangeTimes->{{3.434965591156*^9, 3.434965607399*^9}, {3.434965664946*^9, 3.434965668591*^9}, 3.43496570951*^9}, FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{"F", "[", RowBox[{"x_", ",", "y_", ",", "\[Lambda]_"}], "]"}], " ", "=", " ", RowBox[{ RowBox[{"f", "[", RowBox[{"x", ",", "y"}], "]"}], " ", "+", " ", RowBox[{"\[Lambda]", " ", RowBox[{"g", "[", RowBox[{"x", ",", "y"}], "]"}]}]}]}]], "Input", CellChangeTimes->{{3.43496566991*^9, 3.434965695193*^9}, { 3.4349657350109997`*^9, 3.434965737658*^9}}, FontSize->24], Cell["Then we extremize F by solving some equations:", "Text", CellChangeTimes->{{3.4349657737279997`*^9, 3.4349657968310003`*^9}}, FontSize->24], Cell[BoxData[ RowBox[{"eqnx", " ", "=", " ", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"F", "[", RowBox[{"x", ",", "y", ",", "\[Lambda]"}], "]"}], ",", "x"}], "]"}], "\[Equal]", "0"}]}]], "Input", CellChangeTimes->{{3.434965798245*^9, 3.4349658069370003`*^9}}, FontSize->24], Cell["\<\ Your task: construct eqny and eqn\[Lambda] and then Solve[] for the critical \ points, hopefully confirming the result above.\ \>", "Text", CellChangeTimes->{{3.43496581451*^9, 3.4349658732320004`*^9}}, FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell["Homework task #1", "Subsubsection", CellChangeTimes->{{3.4349659710620003`*^9, 3.434965975619*^9}}, FontSize->24], Cell["\<\ Your task: reproduce figure 3.1 on page 59 of BTM. In addition, decorate \ your plot with a ListPlot of the critical points. PRINT THE RESULTING CELL and hand in as part of the homework. (More precisely, hold off priting until the 2nd task below) As an example, here is our ContourPlot above together with a couple of points \ added:\ \>", "Text", CellChangeTimes->{{3.434965980927*^9, 3.434965989801*^9}, {3.434966111217*^9, 3.434966141789*^9}, {3.434966190484*^9, 3.4349662084519997`*^9}, { 3.434966351309*^9, 3.4349664571429996`*^9}, {3.434966631251*^9, 3.434966687692*^9}}, FontSize->24], Cell[BoxData[ RowBox[{"p4", " ", "=", " ", RowBox[{"ListPlot", "[", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", RowBox[{"-", "1"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "}"}], RowBox[{"5", "/", RowBox[{"Sqrt", "[", "2", "]"}]}]}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", ".03", "]"}], ",", RowBox[{"Hue", "[", "1", "]"}]}], "}"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.434966479051*^9, 3.43496653408*^9}, {3.434966566468*^9, 3.434966603333*^9}}, FontSize->24], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"p2", ",", "p3", ",", "p4"}], "]"}]], "Input", CellChangeTimes->{{3.4349664729960003`*^9, 3.4349664761879997`*^9}, { 3.4349665192469997`*^9, 3.434966544915*^9}}, FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell["Cycloid", "Subsubsection", CellChangeTimes->{{3.434966717082*^9, 3.4349667299309998`*^9}}, FontSize->24], Cell["\<\ Morin asks about the path of a pebble in a tire. For a tire of radius R \ moving at speed v0 we have:\ \>", "Text", CellChangeTimes->{{3.4349667341070004`*^9, 3.434966747153*^9}, { 3.43496683291*^9, 3.4349668681809998`*^9}, 3.434966906417*^9}, FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{"point", "[", "t_", "]"}], " ", "=", " ", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"v0", " ", "t"}], ",", " ", "R"}], "}"}], " ", "+", " ", RowBox[{"R", " ", RowBox[{"{", RowBox[{ RowBox[{"Sin", "[", RowBox[{"v0", " ", RowBox[{"t", "/", " ", "R"}]}], "]"}], ",", RowBox[{"Cos", "[", RowBox[{"v0", " ", RowBox[{"t", "/", "R"}]}], "]"}]}], "}"}]}]}]}]], "Input", CellChangeTimes->{{3.434966908756*^9, 3.434966987316*^9}, { 3.4349670857869997`*^9, 3.4349670860690002`*^9}}, FontSize->24], Cell["\<\ Notice how we are adding vectors here. The first vector is the location of \ the center of the circle, and the 2nd one is the displacement from the center \ to the rim. Your first task: make a ParametricPlot with the numerical choice of say\ \>", "Text", CellChangeTimes->{{3.4349671724110003`*^9, 3.434967307524*^9}}, FontSize->24], Cell[BoxData[ RowBox[{"p", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{"R", "\[Rule]", "1"}], ",", RowBox[{"v0", "\[Rule]", "1"}]}], "}"}]}]], "Input", CellChangeTimes->{{3.434967277333*^9, 3.434967284439*^9}}, FontSize->24], Cell["\<\ Notice that we can get the velocity and acceleration vectors really easily as:\ \>", "Text", CellChangeTimes->{{3.434967325592*^9, 3.4349673258310003`*^9}, { 3.4349673847019997`*^9, 3.434967415769*^9}}, FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{"point", "''"}], "[", "t", "]"}]], "Input", CellChangeTimes->{{3.434967135487*^9, 3.434967153907*^9}, {3.434967419174*^9, 3.434967423358*^9}}, FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{"point", "''"}], "[", "0", "]"}]], "Input", CellChangeTimes->{{3.4350610222650003`*^9, 3.435061024398*^9}}, FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{"point", "'"}], "[", "0", "]"}]], "Input", CellChangeTimes->{{3.435061026476*^9, 3.435061029624*^9}}, FontSize->24], Cell["\<\ The book asks you to find the \"radius of curvature\" at a particular point \ (the peak, to be precise). That is, at any given moment in time we compute the velocity and acceleration \ and then ask: if we were moving in a circle with the same velocity and acceleration, what \ radius would it have?\ \>", "Text", CellChangeTimes->{{3.4349675024700003`*^9, 3.434967558543*^9}, { 3.4349676196619997`*^9, 3.434967619804*^9}, {3.434967656623*^9, 3.434967738476*^9}}, FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell["Homework task #2", "Subsubsection", CellChangeTimes->{{3.434967756454*^9, 3.4349677674519997`*^9}}, FontSize->24], Cell["\<\ Draw a circle of appropriate radius and center on top of the cycloid \ parametric plot. PRINT and hand in with the homework.\ \>", "Text", CellChangeTimes->{{3.434967771793*^9, 3.434967811151*^9}, { 3.4349678946549997`*^9, 3.434967924604*^9}}, FontSize->24] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Extras", "Section", CellChangeTimes->{{3.43496794601*^9, 3.434967947394*^9}}, FontSize->24], Cell[CellGroupData[{ Cell["VectorFieldPlots", "Subsubsection", CellChangeTimes->{{3.4349689613389997`*^9, 3.434968975343*^9}}, FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{"GradientFieldPlot", "[", RowBox[{"f_", ",", RowBox[{"{", RowBox[{"x_", ",", "xmin_", ",", "xmax_"}], "}"}], ",", RowBox[{"{", RowBox[{"y_", ",", "ymin_", ",", "ymax_"}], "}"}], ",", RowBox[{"opts", ":", RowBox[{"OptionsPattern", "[", "]"}]}]}], "]"}], ":=", RowBox[{"VectorPlot", "[", RowBox[{ RowBox[{"Evaluate", "[", RowBox[{"D", "[", RowBox[{"f", ",", RowBox[{"{", RowBox[{"{", RowBox[{"x", ",", "y"}], "}"}], "}"}]}], "]"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "xmin", ",", "xmax"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", "ymin", ",", "ymax"}], "}"}], ",", "opts"}], "]"}]}]], "Input", CellID->587843217], Cell[BoxData[ RowBox[{"GradientFieldPlot", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], " ", "-", " ", RowBox[{"x", " ", "y"}], " ", "+", " ", RowBox[{"y", "^", "2"}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "3"}], ",", "3"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.4663805866157*^9, 3.4663806215557003`*^9}}], Cell["\<\ This defines e.g. GradientFieldPlot, which you can add to the ContourPlot \ above. Also, the function ListVectorFieldPlot can be used to add individual little \ vectors to a plot. Your challenge: decorate your cycloid plot with velocity and acceleration \ vectors for a couple of selected points (I'd choose the peak as one of the points, and a generic \ point for another.)\ \>", "Text", CellChangeTimes->{{3.4349680306070004`*^9, 3.434968031231*^9}, { 3.434968065567*^9, 3.4349681314040003`*^9}, {3.434968190781*^9, 3.434968307243*^9}}, FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell["Rocket Science", "Subsubsection", CellChangeTimes->{{3.434968983065*^9, 3.434968992171*^9}}, FontSize->24], Cell["\<\ In doing the boxcar problem on the homework, you will encounter a \ differential equation:\ \>", "Text", CellChangeTimes->{{3.4349683255959997`*^9, 3.434968371921*^9}, { 3.434968546499*^9, 3.434968559422*^9}, {3.434968946936*^9, 3.434968954979*^9}}, FontSize->24], Cell[BoxData[ RowBox[{"diffeq", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"M0", " ", "+", " ", RowBox[{"b", " ", "t"}]}], ")"}], " ", RowBox[{ RowBox[{"v", "'"}], "[", "t", "]"}]}], " ", "+", " ", RowBox[{"b", " ", RowBox[{"v", "[", "t", "]"}]}]}], " ", "\[Equal]", " ", "F"}]}]], "Input", CellChangeTimes->{{3.4349685679309998`*^9, 3.4349685846610003`*^9}}, FontSize->24], Cell["\<\ While we are here, go ahead and DSolve with appropriate boundary conditions \ (at rest at time t=0) to find the velocity, then Integrate to get the position x[t]. 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