{VERSION 1 0 "X11/Motif" "1.0"}{GLOBALS 3 1}{FONT 0 "-adobe-couri er-medium-r-normal--14-*" "courier" "Courier" 4 14 192 "Courier-B old" 12}{FONT 1 "-adobe-courier-bold-o-normal--14-*" "courier" "C ourier-BoldOblique" 8 14 224 "Courier-BoldOb" 12}{FONT 2 "-adobe- helvetica-bold-r-normal--12-*" "helvetica" "Helvetica-Bold" 8 12 0 "Helvetica" 10}{SCP_R 1 0 91{COM_R 2 0{TEXT 1 1968 "\015\015 SOLVING DIFFERENTIAL EQUATIONS SYMBOLICALLY\015\015 WITH DSOLVE\015\015W. E. Boyce\015Rensselaer Polytechni c Institute\01504/15/94\015\015PURPOSE: To illustrate the use of dsolve to find exact solutions of many differential equations or systems of equations, with or without initial conditions.\015\01 5\015NOTE. Maple has a powerful symbolic differential equation s olver called dsolve. This command directs Maple to seek an exact symbolic expression for the solution of a given differential equ ation, or a system of differential equations, with or without ini tial conditions. Of course, many differential equations do not h ave such exact symbolic solutions, at least not in terms of the u sual well-known functions, so for these equations other Maple com mands may be more helpful. Nevertheless, dsolve works very well for many linear equations and for some nonlinear ones. The exam ples below indicate some of the possibilities for using dsolve e ffectively.\015\015NOTE. dsolve has options that request Maple t o use Laplace transforms, power series, or a numerical algorithm, respectively.\015The laplace option is particularly useful for c ertain initial value problems having a discontinuous or impulsive forcing function. The series and numeric options are most usefu l in those cases where an exact symbolic solution is not availabl e. We do not discuss any of these options here.\015\015NOTE. Fu rther information about dsolve can be obtained by issuing the c ommand `?dsolve'.\015\015 \015\015IMPORTANT NOTE: d solve insists that coefficients be entered as rational, rather th an floating point, numbers. For example, you should enter a coef ficient as 1/2 or 1/4, rather than 0.5 or 0.2.\015\015\015\015\01 5EXAMPLE 1. A FIRST ORDER LINEAR EQUATION\015\015Find the gener al solution of the differential equation\015\015 dy/dt = -2 + sin(t) - (1/4)y.\015\015Also find the solution that satisf ies the initial condition \015y(0) = 2.\015\015\015SOLUTION. The first step is to enter the differential equation. \015"}}{INP_R 3 0 "> "{TEXT 0 45 "eq1 := diff(y(t),t) = -2 + sin(t) - 1/4*y(t) ;"}}{OUT_R 4 0 3{DAG :3n3\`eq1`=3(3n4\`diff`,3(3n3\`y`,2n3\`t`pD+ 7i2x0002j2x0001(3n3\`sin`pCp13p9/3i2x0001j2x0004}}{COM_R 5 0{TEXT 1 211 "Now dsolve can be used to find the general solution of th is equation. In its simplest form the command has two arguments; the first is the equation to be solved and the second is the var iable to be solved for.\015"}}{INP_R 6 0 "> "{TEXT 0 17 "dsolve(e q1,y(t));"}}{OUT_R 7 0 6{DAG =3(3n3\`y`,2n3\`t`+9i2x0008j2x0001(3 n3\`cos`p4/3i2x0016j2x0017(3n3\`sin`p4/3j2x0004p13*5(3n3\`exp`,2+ 3p5/3i2x0001p1ApAn3\`_C1`pApA}}{COM_R 8 0{TEXT 1 0 ""}}{COM_R 9 0 {TEXT 1 423 "This is the general solution of the given differenti al equation. Note that Maple writes the arbitrary constant as _C 1, and that in this case the constant follows the term exp(-t/4) that it multiplies.\015\015To find the solution that also satis fies the initial condition, we include the initial condition in t he dsolve command, using braces to delimit the problem to be so lved. Also we assign the name sol to the result.\015"}}{INP_R 10 0 "> "{TEXT 0 33 "sol := dsolve(\{eq1,y(0)=2\},y(t));"}}{OUT_R 11 0 10{DAG :3n3\`sol`=3(3n3\`y`,2n3\`t`+9i2x0008j2x0001(3n3\`co s`p7/3i2x0016j2x0017(3n3\`sin`p7/3j2x0004p16(3n3\`exp`,2+3p8/3i2x 0001p1D/3j2x0186p16}}{SEP_R 12 0}{COM_R 13 0{TEXT 1 117 "We can n ow assign a name, if we wish, to the expression for y(t) on the right side of the last result, as follows:\015"}}{INP_R 14 0 "> "{TEXT 0 15 "y1 := rhs(sol);"}}{OUT_R 15 0 14{DAG :3n3\`y1`+9i2x0 008j2x0001(3n3\`cos`,2n3\`t`/3i2x0016j2x0017(3n3\`sin`pB/3j2x0004 p11(3n3\`exp`,2+3pC/3i2x0001p18/3j2x0186p11}}{SEP_R 16 0}{COM_R 17 0{TEXT 1 193 "This enables us to perform operations on the sol ution y1 of the initial value problem. For instance, we can eval uate y1 at a given value of t, or we can plot y1 on some give n t interval.\015"}}{INP_R 18 0 "> "{TEXT 0 20 "evalf(subs(t=4,y 1));"}}{OUT_R 19 0 18{DAG e3i4x003537843293i2x0009}}{SEP_R 20 0} {INP_R 21 0 "> "{TEXT 0 34 "plot(y1,t=0..20,title=`Figure 1`);"}} {SEP_R 22 0}{COM_R 23 0{TEXT 1 1 "\015"}}{GRH_R 24 0{RASTER 398 324 1 50 000000000000000000000000000000000000000000F83300000000C0 00000000000000000000000000000000000000000000000000000000000000000 000000000000000000000183000000000F0000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000180000000 000C0000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000018305CC66C1EC00000000000000000000000000 00000000000000000000000000000000000000000000000000000000000001830 76C67C33C00000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000F83163C60C33C000000000000000000000 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00000000000000000000000000008001000030000000000000000E00000000200 00000000000000080C80000000000000000000000000000000000000000000080 01000038000000000000001C0000000060000000000000000080C800000000000 000000000000000000000000000000000000300001C0000000000000018000000 0030000000000000000080C800000000000000000000000000000000000000000 000000700000E00000000000000300000000038000000000000000000C7000000 00000000000000000000000000000000000000000E00000700000000000000700 00000001C000000000000000000C0000000000000000000000000000000000000 00000000001C00800300000000000000E0000000000C000000000000000000C00 0000000000000000000000000000000000000000000007000E001000000000000 00C0000000000E000000000000000000C00000000000000000000000000000000 000000000000000E0007800000000000000008001000000070000000000000000 00C00000000000000000000000000000000000000000000000C0033E000000000 0000000800100000003000000000000000000C000000000000000000000000000 0000000000000000000080FF0F000000000000000000030000800100000000000 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A SECOND ORDER LINEAR HO MOGENEOUS EQUATION\015\015Solve the equation\015\015 y'' - 2y' - 4y = 0.\015\015Find the general solution and also t he solution that satisfies the initial conditions y(0) = 2, y'( 0) = -7/4.\015\015\015SOLUTION: Again the first step is to enter the differential equation. In the following command note that t he first term the second derivative is entered using t$2. This c auses the differentiation operation to be executed twice.\015"}} {INP_R 26 0 "> "{TEXT 0 52 "eq2 := diff(y(t),t$2) - 2*diff(y(t),t ) - 4*y(t) = 0;"}}{OUT_R 27 0 26{DAG :3n3\`eq2`=3+7(3n4\`diff`,3( 3p6,3(3n3\`y`,2n3\`t`p11p11j2x0001pAi2x0002pDi2x0004j2x0000}} {SEP_R 28 0}{INP_R 29 0 "> "{TEXT 0 17 "dsolve(eq2,y(t));"}} {OUT_R 30 0 29{DAG =3(3n3\`y`,2n3\`t`+5*5n3\`_C1`j2x0001(3n3\`exp `,2*5+5pBpB*3j2x0005/3pBj2x0002pBpBp5pBpBpB*5n3\`_C2`pB(3pE,2+3*5 +5i2x0001pBp15pBpBp5pBp2CpBpB}}{SEP_R 31 0}{COM_R 32 0{TEXT 1 444 "This is the general solution. Note that it contains two expone ntial terms, each multiplied by an arbitrary constant. One of th e exponentials is positive, and so grows with increasing t, while the other is negative, and decays as t increases.\015\015Now we solve the initial value problem. To enter the second initial con dition, denote y'(0) by D(y)(0). We will assign the name y2 to this solution, combining the dsolve and rhs commands.\015 "}}{INP_R 33 0 "> "{TEXT 0 52 "y2 := rhs(dsolve(\{eq2,y(0) = 2,D( y)(0)=-7/4\},y(t)));"}}{OUT_R 34 0 33{DAG :3n3\`y2`+5*5+5*3j2x000 5/3j2x0001j2x0002/3i2x0003j2x0008pApApA(3n3\`exp`,2*5+5pApAp6pApA n3\`t`pApApA*5+5p6/3j2x0003p11pApApA(3p17,2+3*5+5i2x0001pAp6pApAp 21pAp36pApA}}{SEP_R 35 0}{COM_R 36 0{TEXT 1 35 "To see the graph of this solution:\015"}}{INP_R 37 0 "> "{TEXT 0 40 "plot(y2,t=0.. 2,y=0..3,title=`Figure 2`);"}}{SEP_R 38 0}{COM_R 39 0{TEXT 1 1 "\ 015"}}{GRH_R 40 0{RASTER 398 361 1 50 000000000000000000000000000 000000000000000F83300000000F0010000000000000000000000000000000000 00000000000000000000000000000000000000000000000000183000000000180 30000000000000000000000000000000000000000000000000000000000000000 00000000000000000000180000000000180300000000000000000000000000000 000000000000000000000000000000000000000000000000000000018305CC66C 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00000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000002000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000 0F000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000020000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000 00000020000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000200000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000 000000000002001000000000000000000000000000000000000000000000000}} {COM_R 41 0{TEXT 1 466 "\015\015Observe that the positive exponen tial term causes the solution to grow rapidly.\015\015\015\015\01 5\015EXAMPLE 3. A SECOND ORDER LINEAR NONHOMOGENEOUS EQUATION\01 5\015Find the general solution of the equation\015\015 y'' + 4 y' + 5 y = 3 exp(-t) + 4 sin(3t).\015\015Also find the solution satisfying the initial conditions\015\015 y(0) = 3/2, y'(0) = 2,\015\015and plot its graph. \015\015\015SOLUTION. As in the preceding examples we find the general solution by entering the e quation and then using dsolve.\015 "}}{INP_R 42 0 "> "{TEXT 0 65 "eq3 := diff(y(t),t$2)+4*diff(y(t),t)+5*y(t)=3*exp(-t)+4*sin(3*t );"}}{OUT_R 43 0 42{DAG :3n3\`eq3`=3+7(3n4\`diff`,3(3p6,3(3n3\`y` ,2n3\`t`p11p11j2x0001pAj2x0004pDj2x0005+5(3n3\`exp`,2+3p11i2x0001 j2x0003(3n3\`sin`,2+3p11p26p18}}{SEP_R 44 0}{INP_R 45 0 "> "{TEXT 0 24 "sol := dsolve(eq3,y(t));"}}{OUT_R 46 0 45{DAG :3n3\`sol`=3 (3n3\`y`,2n3\`t`+19(3n3\`exp`,2+3p8i2x0001/3j2x0003j2x0002*5(3n3\ `sin`p7j2x0001(3n3\`cos`,2+3p8j2x0004p1D/3i2x0002j2x0005*5p19p1D( 3p1Ap22p1D/3p1Dp2B*5p19p1D(3p20,2+3p8p16p1D/3p11p16*5p19p1D(3p1Ap 3Cp1D/3p1Dp16*3(3p20p7p14/3i2x0013p2B*5p4Fp16p19p1Dp53p4F/3j2x000 7j2x0010*3p4Fp2B/3j2x0008p2B*5p4Fp25p19p1D/3j2x0016p2B*7n3\`_C1`p 1D(3pC,2+3p8p29p1Dp19p1Dp1D*7n3\`_C2`p1Dp77p1Dp4Fp1Dp1D}}{SEP_R 47 0}{COM_R 48 0{TEXT 1 582 "The two terms that involve -C1 and _ C2 form the general solution of the homogeneous equation, while t he other terms are a particular solution of the nonhomogeneous eq uation. The term in sol that contains exp(-t) arises from the co rresponding term in the forcing function. The remaining trigonom etric terms in sol come from the sin(3t) term in the forcing func tion. If we solved this equation by hand we would expect to get only two trigonometric terms, one in sin(3t) and the other in cos (3t). We can cause Maple to write the solution in this simpler f orm in the following way.\015"}}{INP_R 49 0 "> "{TEXT 0 18 "combi ne(sol,trig);"}}{OUT_R 50 0 49{DAG =3(3n3\`y`,2n3\`t`+B(3n3\`exp` ,2+3p5i2x0001/3j2x0003j2x0002(3n3\`sin`,2+3p5p11/3pEj2x0010(3n3\` cos`p18/3i2x0003p1E*7n3\`_C1`j2x0001(3p9,2+3p5i2x0002p2B(3p16p4p2 Bp2B*7n3\`_C2`p2Bp2Dp2B(3p21p4p2Bp2B}}{SEP_R 51 0}{COM_R 52 0 {TEXT 1 158 "This is the form of the solution that we would obtai n by hand, using\015the method of undetermined coefficients.\015\ 015Now let us use the initial conditions as well.\015"}}{INP_R 53 0 "> "{TEXT 0 32 "init := y(0) = 3/2, D(y)(0) = 2;"}}{OUT_R 54 0 53{DAG :3n4\`init`,3=3(3n3\`y`,2j2x0000/3j2x0003j2x0002=3(3(3n3\ `D`,2p7p9pF}}{SEP_R 55 0}{INP_R 56 0 "> "{TEXT 0 31 "sol := dsolv e(\{eq3,init\},y(t)):"}}{SEP_R 57 0}{INP_R 58 0 "> "{TEXT 0 29 "y 3 := combine(rhs(sol),trig);"}}{OUT_R 59 0 58{DAG :3n3\`y3`+B(3n3 \`exp`,2+3n3\`t`i2x0001/3j2x0003j2x0002(3n3\`sin`,2+3p9pE/3pBj2x0 010(3n3\`cos`p15/3i2x0003p1B*5(3p5,2+3p9i2x0002j2x0001(3p13,2p9p2 D/3j2x0022j2x0005*5p26p2D(3p1Ep31p2D/3pEp1B}}{SEP_R 60 0}{COM_R 61 0{TEXT 1 668 "This is the solution to the given initial value problem. Note that values have been assigned to the arbitrary c onstants that appeared in the previous expression for y(t). \015\ 015What should we expect its graph to look like? The second and third terms depend only on sin(3t) and cos(3t), so they represent a steady oscillation of frequency 3, or of period 2Pi/3. The ot her terms all include a negative exponential factor, so they cont ribute significantly only when t is small, and then tend to zero as t becomes large. The graph then is a somewhat complicated cu rve for small t, but approaches a simple oscillation as t increas es. To confirm this, we ask for a plot.\015"}}{INP_R 62 0 "> " {TEXT 0 34 "plot(y3,t=0..15,title=`Figure 3`);"}}{SEP_R 63 0} {COM_R 64 0{TEXT 1 3 " \015\015"}}{GRH_R 65 0{RASTER 398 324 1 50 000000000000000000000000000000000000000000F83300000000F001000000 00000000000000000000000000000000000000000000000000000000000000000 00000000000001830000000001803000000000000000000000000000000000000 00000000000000000000000000000000000000000000000018000000000018030 00000000000000000000000000000000000000000000000000000000000000000 00000000000000000018305CC66C1E00030000000000000000000000000000000 00000000000000000000000000000000000000000000000000000183076C67C33 00030000000000000000000000000000000000000000000000000000000000000 00000000000000000000000F83163C60C33E00100000000000000000000000000 0000000000000000000000000000000000000000000000000000000000183063C 60C3F000300000000000000000000000000000000000000000000000000000000 0000000000000000000000000000183063C60C030003000000000000000000000 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00000000000000000000000000000C00000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000C 00000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000C000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000 0000C000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000}}{COM_R 66 0{TEXT 1 418 "\015\015 To appreciate fully what Maple has done on this example, you shou ld think about what you would have had to do to obtain the same r esults by hand.\015\015\015\015\015\015EXAMPLE 4. Find the gener al solution of the equation\015\015 y'''' + y''' + y'' - 9 y' - 10 y = 0,\015\015and also find the solution satisfying the init ial conditions\015\015 y(0) = 1, y'(0) = 2, y''(0) = -1, y'''(0 ) = -3.\015\015\015SOLUTION. We proceed as in the preceding exam ples.\015"}}{INP_R 67 0 "> "{TEXT 0 87 "eq4 := diff(y(t),t$4) + d iff(y(t),t$3) + diff(y(t),t$2) - 9*diff(y(t),t) - 10*y(t) = 0;"}} {COM_R 68 0{TEXT 1 2 "\015\015"}}{OUT_R 69 0 67{DAG :3n3\`eq4`=3+ B(3n4\`diff`,3(3p6,3(3p6,3(3p6,3(3n3\`y`,2n3\`t`p17p17p17p17j2x00 01pAp1DpDp1Dp10i2x0009p13i2x0010j2x0000}}{SEP_R 70 0}{INP_R 71 0 "> "{TEXT 0 17 "dsolve(eq4,y(t));"}}{OUT_R 72 0 71{DAG =3(3n3\`y` ,2n3\`t`+9*5n3\`_C1`j2x0001(3n3\`exp`,2+3p5j2x0002pBpB*5n3\`_C2`p B(3pE,2+3p5i2x0001pBpB*7n3\`_C3`pBp1BpB(3n3\`sin`p10pBpB*7n3\`_C4 `pBp1BpB(3n3\`cos`p10pBpB}}{SEP_R 73 0}{COM_R 74 0{TEXT 1 366 "Th is is the general solution. Its form indicates that two roots of the characteristic polynomial are real (2 and -1) and that the o ther two are complex conjugates (-1 + 2I and -1 -2I).\015\015Now we enter the initial conditions. Note that the second derivative is expressed by (D@@2) and similarly for the third derivative. The parentheses around (D@@2) are essential.\015"}}{INP_R 75 0 "> "{TEXT 0 60 "init := y(0)=1, D(y)(0)=2, (D@@2)(y)(0)=-1, (D@@3)( y)(0)=-3;"}}{OUT_R 76 0 75{DAG :3n4\`init`,5=3(3n3\`y`,2j2x0000j2 x0001=3(3(3n3\`D`,2p7p9j2x0002=3(3(3(3n3\`@@`,3p11p16p13p9i2x0001 =3(3(3(3p1C,3p11j2x0003p13p9i2x0003}}{SEP_R 77 0}{INP_R 78 0 "> " {TEXT 0 35 "y4 := rhs(dsolve(\{eq4,init\},y(t)));"}}{OUT_R 79 0 78{DAG :3n3\`y4`+9(3n3\`exp`,2+3n3\`t`j2x0002/3j2x0001j2x0003(3p5 ,2+3p9i2x0001/3j2x0011j2x0012*5p12pE(3n3\`sin`p7pEpE*5p12pE(3n3\` cos`p7pE/3p17j2x0004}}{SEP_R 80 0}{COM_R 81 0{TEXT 1 330 "What do you think the graph of this solution looks like? Once you have formed an opinion, ask Maple for a plot to confirm your conclusio n.\015\015\015\015EXAMPLE 5. Find the general solution of the sy stem\015\015 dx/dt = -2 x + y, dy/dt = x - 2 y.\015\015Also fi nd the solution that satisfies the initial conditions x(0) = 0, y (0) = 2.\015\015\015\015SOLUTION.\015"}}{INP_R 82 0 "> "{TEXT 0 65 "sys := diff(x(t),t) = -2*x(t)+y(t), diff(y(t),t) = x(t) - 2*y (t);"}}{OUT_R 83 0 82{DAG :3n3\`sys`,3=3(3n4\`diff`,3(3n3\`x`,2n3 \`t`pE+5pAi2x0002(3n3\`y`pDj2x0001=3(3p6,3p15pE+5pAp19p15p13}} {COM_R 84 0{TEXT 1 0 ""}}{INP_R 85 0 "> "{TEXT 0 26 "dsolve(\{sys \},\{x(t),y(t)\});"}}{OUT_R 86 0 85{DAG %2,3=3(3n3\`x`,2n3\`t`+5* 5n3\`_C1`j2x0001(3n3\`exp`,2+3p7i2x0001pDpD*5n3\`_C2`pD(3p10,2+3p 7i2x0003pDp15=3(3n3\`y`p6+5pApDp19pD}}{COM_R 87 0{TEXT 1 0 ""}} {COM_R 88 0{TEXT 1 221 "Observe that both x(t) and y(t) are linea r combinations of the same two exponential functions, and that th ere are only two distinct arbitrary constants in the solution.\01 5\015Here is the solution of the initial value problem.\015"}} {INP_R 89 0 "> "{TEXT 0 40 "dsolve(\{sys,x(0)=0,y(0)=2\},\{x(t),y (t)\});"}}{OUT_R 90 0 89{DAG %2,3=3(3n3\`x`,2n3\`t`+5(3n3\`exp`,2 +3p7i2x0001j2x0001(3pB,2+3p7i2x0003p10=3(3n3\`y`p6+5pAp12p14p12}} {COM_R 91 0{TEXT 1 0 ""}}{COM_R 92 0{TEXT 1 1707 "\015\015NOTE: Solving a linear homogeneous equation with constant coefficients, or a system of such equations, of higher than second order is no t necessarily a simple problem, even for Maple. The main mathema tical task is to find the roots of the characteristic polynomial. Equations for which the roots are rational (real or complex) ar e handled with relative ease by Maple, but other equations may le ad to very slow response times, or may even cause the system to f ail. This reflects the fact that such problems may be very diffi cult. Some of these more difficult problems are explored in anoth er worksheet.\015\015\015\015\015PROBLEMS\015\015In each of the f ollowing problems use dsolve to find the general solution of the given differential equation. If initial conditions are given, al so find the solution that satisfies them. Plot this solution.\015 \015 1. dy/dt = t + exp(-2t) + 3y, y(0) = 2\015\015 2. dy/dt = 3 sin(t) + 1 - y, y(0) = -1\015\015 3. y'' + (5/2)y' + y = 0, y(0) = 2, y'(0) = 3\015\015 4. y'' + 6y' + 10y = 0, y(0) = 1, y'(0) = 1\015\015 5. y'' + (5/2)y' + y = 3 cos(2t) + 6 exp(-2 t),\015\015 y(0) = 2, y'(0) = 3\015\015 6. y'' + 6y' + 10y \015 \015 = 5 exp(-3t)cos(t) + 2 exp(-t)cos(3t),\ 015\015 y(0) = 1, y'(0) = 2\015\015 7. y''' + 3y'' + 3y' + 2y = 0, \015\015 y(0) = 2, y'(0) = -1, y''(0) = 3\015\01 5 8. y'''' + 4y''' + 10y'' + 11y' + 10y = 0,\015\015 y(0) = 2, y'(0) = 3, y''(0) = -1, y'''(0) = 0\015\015 9. y''' + 3y'' + 3y' + 2y \015\015 = exp(-t)sin(t) + 3t^2 - 4t + 5,\015\ 015 y(0) = 2, y'(0) = -1, y''(0) = 3\015\01510. dx/dt = 3x - 2y, dy/dt = 2x - 2y, \015\015 x(0) = 3, y(0) = 2\015\ 01511. dx/dt = x - 5y, dy/dt = x - 3y, \015\015 x(0) = 3, y(0) = 2\015\01512. dx/dt = 2x - 5y, dy/dt = x - 2y, \015\ 015 x(0) = 3, y(0) = 2\015\015\015"}}}{END}