{VERSION 1 0 "X11/Motif" "1.0"}{GLOBALS 3 1}{FONT 0 "-adobe-couri er-bold-r-normal--14-*" "courier" "Courier-Bold" 8 14 192 "Courie r-Bold" 12}{FONT 1 "-adobe-courier-bold-o-normal--14-*" "courier" "Courier-BoldOblique" 8 14 224 "Courier-BoldOb" 12}{FONT 2 "-ado be-courier-medium-r-normal--14-*" "courier" "Courier" 4 14 192 "C ourier" 12}{SCP_R 1 0 54{COM_R 2 0{TEXT 1 684 " \015 A FIRST ORDER LINEAR DIFFERENTIAL EQUATION\015\015W. E. Boyce\015Renssela er Polytechnic Institute\01504/15/94\015\015PURPOSE: To show how Maple can be used to investigate a first order linear differenti al equation, using both graphical and analytical methods.\015\015 EXAMPLE. Consider the first order linear differential equation\0 15\015 dy/dt = - 2 + t - y.\015\015(a) Draw the directio n field for this equation and plot several solutions.\015\015(b) Also solve the equation analytically and find the solution that satisfies the initial condition y(0) = 0.\015\015SOLUTION. We w ill use the Maple command `DEplot' to draw the direction field. In order to use this command we first call up the DEtools package .\015"}}{INP_R 3 0 "> "{TEXT 0 14 "with(DEtools);"}}{OUT_R 4 0 3 {DAG [2,8n4\`DEplot`n4\`DEplot1`n4\`DEplot2`n5\`Dchangevar`n4\`PD Eplot`n5\`dfieldplot`n6\`phaseportrait`}}{SEP_R 5 0}{COM_R 6 0 {TEXT 1 448 "\015This command loads the DEtools package and lists the commands that it contains.\015\015Information about any of t hese commands can be obtained online through Maple's help facilit y. For example, the command ?DEplot will produce information a bout DEplot.\015\015(a) The first step in solving the problem is to enter the differential equation. This can be done in severa l ways, as described in the help screen. We will do it as shown in the next command.\015"}}{INP_R 7 0 "> "{TEXT 0 37 "eq1 := diff (y(t),t) = - 2 + t - y(t);"}}{OUT_R 8 0 7{DAG :3n3\`eq1`=3(3n4\`d iff`,3(3n3\`y`,2n3\`t`pD+7i2x0002j2x0001pDp13p9i2x0001}}{SEP_R 9 0}{COM_R 10 0{TEXT 1 569 "\015The following DEplot command will p roduce a direction field for this equation. The first argument i s the name of the equation, the second is a list (enclosed in squ are brackets) of the variables with the independent variable firs t, and the third and fourth are the horizontal and vertical dimen sions of the rectangle in which the direction field will be drawn . The `arrows' option directs Maple to draw a direction field us ing THIN arrows. (It is also possible to request SLIM or THICK a rrows.) The title option is not essential but can be used to lab el the plot.\015"}}{INP_R 11 0 "> "{TEXT 0 62 "DEplot(eq1,[t,y],t =0..6,y=-2..2,arrows=THIN,title=`Figure 1`);"}}{SEP_R 12 0}{COM_R 13 0{TEXT 1 2 " \015"}}{GRH_R 14 0{RASTER 398 323 1 50 000000000 000000000000000000000000000000000F8330000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000 000183000000000C0000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000180000000000F00000000000000 00000000000000000000000000000000000000000000000000000000000000000 0000000018305CC66C1EC00000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000183076C67C33C000000000 00000000000000000000000000000000000000000000000000000000000000000 0000000000000F83163C60C33C000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000183063C60C3FC0000 00000000000000000000000000000000000000000000000000000000000000000 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Furthermore, the initial point must occur as the fourth argument, that is, between the horizontal an d vertical dimensions of the rectangle. If you want only the sol ution and not the direction field in the background, omit the `ar rows' option.\015\015If you want Maple to choose the vertical dim ension of the rectangle, you can also omit the y specification. For a steeply rising (or falling) curve this may lead to a larg er y interval than you want.\015\015Sometimes you may want to p lot the solution through each of several initial points. One way to do this is to list the coordinates of each point in brackets a nd then to enclose the entire collection of points in braces. Th is would replace the \{[0,0]\} in the last DEplot command. Oft en it is easier to use Maple's `seq' (for sequence) command to pr oduce the list of points.\015"}}{INP_R 21 0 "> "{TEXT 0 31 "init := seq([0,0.5*i],i=-4..4);"}}{OUT_R 22 0 21{DAG :3n4\`init`,A[2,3 j2x0000e3i2x0020i2x0001[2,3p7e3i2x0015pC[2,3p7e3i2x0010pC[2,3p7e3 i2x0005pC[2,3p7p7[2,3p7e3j2x0005pC[2,3p7e3j2x0010pC[2,3p7e3j2x001 5pC[2,3p7e3j2x0020pC}}{COM_R 23 0{TEXT 1 176 "\015This produces a list of nine points, spaced 0.5 apart on the y axis, and names `init'. These points can now be inserted in the DEplot command; note the braces around init.\015"}}{INP_R 24 0 "> "{TEXT 0 69 "D Eplot(eq1,[t,y],t=0..6,\{init\},y=-2..2,arrows=THIN,title=`Figure 3`);"}}{SEP_R 25 0}{COM_R 26 0{TEXT 1 1 "\015"}}{GRH_R 27 0 {RASTER 398 326 1 50 000000000000000000000000000000000000000000F8 33000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000183000000000F0010000000000000000 00000000000000000000000000000000000000000000000000000000000000000 00018000000000098030000000000000000000000000000000000000000000000 0000000000000000000000000000000000000018305CC66C1E180300000000000 00000000000000000000000000000000000000000000000000000000000000000 00000000183076C67C33000300000000000000000000000000000000000000000 0000000000000000000000000000000000000000000F83163C60C33E001000000 00000000000000000000000000000000000000000000000000000000000000000 0000000000000183063C60C3F0003000000000000000000000000000000000000 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The default is a 20 by 20 grid, producing 21 arrows in ea ch row or column. To change this grid insert the option grid=[m ,n] in the DEplot command. This will produce a direction field o n an m by n grid. For example,\015"}}{INP_R 29 0 "> "{TEXT 0 58 "DEplot(eq1,[t,y],t=0..6,y=-2..2,grid=[15,10],arrows=THIN);"} }{COM_R 30 0{TEXT 1 371 "\015produces a direction field on a 15 by 10 grid.\015\015\015ANALYTICAL SOLUTION\015\015To solve the given differential equation symbolically we use the Maple command `dsolve', which is in the standard Maple library (not the DEtool s package).\015\015The general solution is found by the following command; the first argument is the equation and the second is th e variable to be solved for.\015\015"}}{INP_R 31 0 "> "{TEXT 0 17 "dsolve(eq1,y(t));"}}{OUT_R 32 0 31{DAG =3(3n3\`y`,2n3\`t`+7i2x0 003j2x0001p5pA*5(3n3\`exp`,2+3p5i2x0001pAn3\`_C1`pApA}}{SEP_R 33 0}{COM_R 34 0{TEXT 1 0 ""}}{COM_R 35 0{TEXT 1 379 "Note the way i n which Maple designates arbitrary constants. Often, but not alw ays, the constant appears after the term that it multiplies.\015\ 015To find the solution of an initial value problem we include th e initial condition in the statement of the problem, enclosing th e differential equation and initial condition in braces. For lat er reference we will name the solution `sol'.\015"}}{INP_R 36 0 " > "{TEXT 0 33 "sol := dsolve(\{eq1,y(0)=0\},y(t));"}}{OUT_R 37 0 36{DAG :3n3\`sol`=3(3n3\`y`,2n3\`t`+7i2x0003j2x0001p8pD(3n3\`exp` ,2+3p8i2x0001j2x0003}}{SEP_R 38 0}{COM_R 39 0{TEXT 1 362 "\015It is important to understand that `sol' is an equation, not a Maple expression. In order to calculate values of the solution at par ticular values of t, or to plot the solution, or to perform othe r operations on it, we must pick off the right hand side of `sol ' and give it a name, such as y1. In the following command `rhs' stands for `right hand side'. \015"}}{INP_R 40 0 "> "{TEXT 0 15 "y1 := rhs(sol);"}}{OUT_R 41 0 40{DAG :3n3\`y1`+7i2x0003j2x0001n 3\`t`p6(3n3\`exp`,2+3p8i2x0001j2x0003}}{SEP_R 42 0}{COM_R 43 0 {TEXT 1 53 "\015The last two commands can be combined, if you wis h.\015"}}{INP_R 44 0 "> "{TEXT 0 37 "y2 := rhs(dsolve(\{eq1,y(0)= 0\},y(t)));"}}{OUT_R 45 0 44{DAG :3n3\`y2`+7i2x0003j2x0001n3\`t`p 6(3n3\`exp`,2+3p8i2x0001j2x0003}}{SEP_R 46 0}{COM_R 47 0{TEXT 1 105 "\015Now we can evaluate y1 at a particular value of t, or plot its graph, using standard Maple commands.\015"}}{INP_R 48 0 "> "{TEXT 0 20 "evalf(subs(t=3,y1));"}}{OUT_R 49 0 48{DAG e3j4x0 01493612051i2x0010}}{SEP_R 50 0}{INP_R 51 0 "> "{TEXT 0 41 "plot( y1,t=0..6,y=-2..2,title=`Figure 4`);"}}{SEP_R 52 0}{COM_R 53 0 {TEXT 1 1 "\015"}}{GRH_R 54 0{RASTER 398 331 1 50 000000000000000 000000000000000000000000000F8330000000000000000000000000000000000 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symbolic solution for a number of t values, and then drawing a curve through the points generated in this way. The g raph in Figure 2 was produced by evaluating approximate values of y by a numerical process applied directly to the differential equation, without knowing a symbolic expression for the solution. The latter process is more general and is particularly useful f or those equations (which are very numerous) for which analytical solutions cannot be found.\015\015\015\015\015"}}}{END}