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174 chapter3linearsystems exercises for section 3 1 1 since a 0 paul s making a proat x 0 has a beneacial effect on paul s proats in the future because ...

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                 174   CHAPTER3LINEARSYSTEMS
                 EXERCISES FOR SECTION 3.1
                   1. Since a > 0, Paul’s making a proÀt(x > 0) has a beneÀcial effect on Paul’s proÀts in the future
                     because the ax term makes a positive contribution to dx/dt. However, since b < 0, Bob’s making
                     aproÀt(y > 0) hinders Paul’s ability to make proÀt because the by term contributes negatively to
                     dx/dt. Roughlyspeaking, business is good for Paul if his store is proÀtable and Bob’s is not. In fact,
                     since dx/dt = x Š y,Paul’sproÀts will increase whenever his store is more proÀtable than Bob’s.
                        Even though dx/dt = dy/dt = x Š y for this choice of parameters, the interpretation of the
                     equation is exactly the opposite from Bob’s point of view. Since d < 0, Bob’s future proÀts are hurt
                     whenever he is proÀtable because dy < 0. But Bob’s proÀts are helped whenever Paul is proÀtable
                     since cx > 0. Once again, since dy/dt = x Š y, Bob’s proÀts will increase whenever Paul’s store is
                     moreproÀtable than his.
                        Finally, note that both x and y change by identical amounts since dx/dt and dy/dt are always
                     equal.
                   2. Since a = 2, Paul’s making a proÀt(x > 0) has a beneÀcial effect on Paul’s future proÀts because
                     the ax term makes a positive contribution to dx/dt. However, since b =Š1, Bob’s making a proÀt
                     (y > 0) hinders Paul’s ability to make proÀt because the by term contributes negatively to dx/dt.
                     In some sense, Paul’s proÀtability has twice the impact on his proÀts as does Bob’s proÀtability. For
                     example, Paul’s proÀts will increase whenever his proÀts are at least one-half of Bob’s proÀts since
                     dx/dt = 2x Š y.
                        Since c = d = 0, dy/dt = 0. Consequently, Bob’s proÀts are not affected by the proÀtability of
                     either store, and hence his proÀts are constant in this model.
                   3. Since a = 1andb = 0, we have dx/
                                                dt = x. Hence, if Paul is making a proÀt(x > 0), then those
                     proÀts will increase since dx/dt is positive. However, Bob’s proÀts have no effect on Paul’s proÀts.
                     (Note that dx/dt = x is the standard exponential growth model.)
                        Since c = 2andd = 1, proÀts from both stores have a positive effect on Bob’s proÀts. In some
                     sense, Paul’s proÀts have twice the impact of Bob’s proÀts on dy/dt.
                   4. Since a =Š1andb = 2, Paul’s making a proÀt has a negative effect on his future proÀts. However,
                     if Bob makes a proÀt, then Paul’s proÀts beneÀt. Moreover, Bob’s proÀtability has twice the impact
                     as does Paul’s. In fact, since dx/dt =Šx + 2y,Paul’sproÀts will increase if Šx + 2y > 0or,in
                     other words, if Bob’s proÀts are at least one-half of Paul’s proÀts.
                        Since c = 2andd =Š1, Bob is in the same situation as Paul. His proÀts contribute negatively
                     to dy/dt since d =Š1. However, Paul’s proÀtability has twice the positive effect.
                        NotethatthismodelissymmetricinthesensethatbothPaulandBobperceiveeachothersproÀts
                     in the same way. This symmetry comes from the fact that a = d and b = c.
                         x                                                      
                                 dY     21                       x      dY       03
                   5. Y =     ,  dt =         Y6.Y= , dt=                                Y
                           y            11                       y             Š0.33π
                        ⎛ p ⎞    dY   ⎛ 3 Š2 Š7 ⎞
                   7. Y = ⎜  ⎟,     =⎜              ⎟Y
                           q            Š206
                        ⎝    ⎠    dt  ⎝             ⎠
                           r              07.32
                                                                  3.1 Properties of Linear Systems and The Linearity Principle     175
                           8.   dx =Š3x +2πy                                        9.  dx =βy
                                dt                                                       dt
                                dy =4x Š y                                              dy =γx Šy
                                dt                                                       dt
                          10.   (a)              y                 (b)             y                 (c)   x, y
                                               2                                 2                       40     x(t)
                                                                                                                     ❅
                                                                                                         20          ❅❘
                                                                                                                            ❅■
                                                                                                 x                          ❅y(t)
                                                                 x     Š22 t
                                    Š22                                                                                12
                                              Š2                               Š2
                          11.   (a)              y                 (b)             y                 (c)   x, y     x(t)      y(t)
                                               2                                 2                         3             
                                                                                                                ✠        ✠
                                                                                                                       10         20 t
                                                                 x                               x       Š3
                                                                       Š22
                                    Š22
                                              Š2                               Š2
                          12.   (a)              y                 (b)             y                 (c)       y(t)x, y
                                                                                 2                           20
                                               2                                                            ✠
                                                                                                                  10
                                                                                                 x                                 t
                                                                 x                                       Š11
                                                                                                                Š10
                                                                       Š22
                                                                                                             ❅■
                                     Š22 ❅x(t)
                                             Š2                                Š2
                      176     CHAPTER3LINEARSYSTEMS
                        13.   (a)            y               (b)            y                (c)   x, y        y(t)
                                            2                             2                              x(t)
                                                                                                  1    
                                                                                                     ✠     ✠
                                                                                                                           t
                                                                                         x       Š1        123
                                                            x    Š22
                                 Š22
                                          Š2                             Š2
                        14.   (a) If a = 0, then detA = ad Š bc = bc. Thus both b and c are nonzero if detA = 0.
                             (b) Equilibrium points (x , y ) are solutions of the simultaneous system of linear equations
                                                      0  0
                                                                     ⎧
                                                                     ⎨ ax +by =0
                                                                           0     0
                                                                     ⎩ cx +dy =0.
                                                                           0     0
                                 If a = 0, the Àrst equation reduces to by = 0, and since b = 0, y    = 0. In this case, the
                                                                          0                          0
                                 second equation reduces to cx = 0, so x = 0 as well. Therefore, (x , y ) = (0,0) is the only
                                                              0          0                          0  0
                                 equilibrium point for the system.
                        15. ThevectorÀeldatapoint(x ,y )is(ax +by ,cx +dy ),soinorderforapointtobeanequilibrium
                                                       0  0       0    0    0    0
                            point, it must be a solution to the system of simultaneous linear equations
                                                                   ⎧
                                                                   ⎨ ax +by =0
                                                                        0      0
                                                                   ⎩ cx +dy =0.
                                                                        0      0
                                If a = 0, we know that the Àrst equation is satisÀed if and only if
                                                                              b
                                                                      x =Š y .
                                                                       0      a 0
                                Nowweseethat any point that lies on this line x = (Šb/a)y also satisÀes the second linear
                                                                                0            0
                            equation cx + dy = 0. In fact, if we substitute a point of this form into the second component of
                                       0      0
                            the vector Àeld, we have
                                                                           
 b
                                                             cx +dy =c Š           y +dy
                                                               0      0        a    0      0
                                                                          
 bc       
                                                                       = Š +d y
                                                                              a         0
                                                                       =
adŠbcy
                                                                               a       0
                                                                       = detAy
                                                                            a    0
                                                                       =0,
                                                                  3.1 Properties of Linear Systems and The Linearity Principle     177
                              since we are assuming that detA = 0. Hence, the line x = (Šb/a)y consists entirely of equilib-
                              rium points.                                                0              0
                                   If a = 0andb = 0, then the determinant condition detA = ad Š bc = 0 implies that c = 0.
                              Consequently, the vector Àeld at the point (x , y ) is (by ,dy ).Sinceb = 0, we see that we get
                                                                              0   0        0    0
                              equilibrium points if and only if y = 0. In other words, the set of equilibrium points is exactly the
                              x-axis.                             0
                                   Finally, if a = b = 0, then the vector Àeld at the point (x , y ) is (0,cx + dy ). In this case,
                                                                                                 0  0           0      0
                              weseethatapoint(x ,y )isanequilibriumpoint if and only if cx +dy = 0. Since at least one of
                                                     0  0                                           0      0
                              c or d is nonzero, the set of points (x , y ) that satisfy cx + dy = 0 is precisely a line through the
                              origin.                                0   0                 0      0
                          16.   (a) Let v = dy/dt.Thendv/dt = d2y/dt2 =ŠqyŠ p(dy/dt) =ŠqyŠ pv. Thusweobtainthe
                                    system
                                                                             dy =v
                                                                             dt
                                                                             dv =ŠqyŠ pv.
                                                                             dt
                                    In matrix form, this system is written as
                                                                  ⎛ dy ⎞                      
                                                                  ⎜ dt ⎟=            01y .
                                                                  ⎝ dv ⎠           Šq Šp            v
                                                                      dt
                                (b) The determinant of this matrix is q. Hence, if q = 0, we know that the only equilibrium point
                                    is the origin.
                                (c) If y is constant, then v = dy/dt is identically zero. Hence, dv/dt = 0.
                                         Also, the system reduces to
                                                                  ⎛ dy ⎞                      
                                                                  ⎜ dt ⎟=            01y ,
                                                                  ⎝ dv ⎠           Šq Šp            0
                                                                      dt
                                    which implies that dv/dt =Šqy.
                                         Combining these two observations, we obtain dv/dt =Šqy = 0, and if q = 0, then
                                    y = 0.
                          17. The Àrst-order system corresponding to this equation is
                                                                          dy =v
                                                                          dt
                                                                          dv =ŠqyŠ pv.
                                                                          dt
                                (a) If q = 0, then the system becomes           dy
                                                                                dt =v
                                                                                dv =Špv,
                                                                                dt
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...Chapterlinearsystems exercises for section since a paul s making proat x has beneacial effect on proats in the future because ax term makes positive contribution to dx dt however b bob aproat y hinders ability make by contributes negatively roughlyspeaking business is good if his store proatable and not fact sproats will increase whenever more than even though dy this choice of parameters interpretation equation exactly opposite from point view d are hurt he but helped cx once again moreproatable finally note that both change identical amounts always equal some sense proatability twice impact as does example at least one half c consequently affected either hence constant model andb we have then those no standard exponential growth andd stores negative beneat moreover or other words same situation contribute notethatthismodelissymmetricinthesensethatbothpaulandbobperceiveeachothersproats way symmetry comes p q r properties linear systems linearity principle t deta ad bc thus nonzero equ...

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