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5.7进一步的微分方程

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1 1 b 2 2 b 2摄氏度 二维 3 3 b 3 c 4 4 b 4摄氏度 5 5 b 5度 6 6 b 6摄氏度 7一个 7 b 7 c

问题1

标志:6

使用步长为0.1的欧拉方法来找到的值的近似值x{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px">而且y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px">当t=0.5{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="T = 0.5" style="vertical-align:-6px;height:22px;width:47px">对于给定初始条件下的下列耦合微分方程组:

(一)
dxdt=x2+4ty{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分子dx除以分母dt端分数等于x方加4t y" style="vertical-align:-17px;height:47px;width:103px">

dydt=-3x+y-t{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分子dy除以分母dt末端分数等于- 3x + y - t" style="vertical-align:-17px;height:47px;width:131px">

x=2, y=1  when  t=0{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="X = 2 y = 1空间空间w h ne空间空间t = 0" style="vertical-align:-6px;height:22px;width:174px">

    评估你的分数
    关键概念
    欧拉方法:耦合系统

    问题1 b

    标志:6
    (b)


    x˙=-x+e-ty{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="上面有点的X等于- X加上e的- t次方y次方" style="vertical-align:-6px;height:24px;width:105px" loading="lazy">

    y˙=e-tx+y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="Y上点乘等于e的- t次方(x + Y" style="vertical-align:-6px;height:24px;width:88px" loading="lazy">

    x=1, y=-1  when  t=0{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="X = 1 y = - 1空间空间w h ne空间空间t = 0" style="vertical-align:-6px;height:22px;width:191px" loading="lazy">

      评估你的分数
      关键概念
      欧拉方法:耦合系统

      问题2

      标志:6

      考虑下面的微分方程组:

      dxdt=x+2y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分子dx除以分母dt端分数等于x + 2y" style="vertical-align:-17px;height:47px;width:91px" loading="lazy">

      dydt=-3x-4y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分子dy除以分母dt端分数等于- 3x - 4y" style="vertical-align:-17px;height:47px;width:117px" loading="lazy">

      (一)
      找出矩阵的特征值和相应的特征向量12-3-4{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="开括号表第1行2行cell - 3结束cell - 4结束cell结束表闭括号" style="vertical-align:-21px;height:54px;width:75px" loading="lazy">．
        评估你的分数

        问题2 b

        分数:2
        (b)
        因此，写出方程组的通解。
          评估你的分数

          问题2摄氏度

          分数:3

          当t=0{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="T = 0" style="vertical-align:-6px;height:22px;width:33px" loading="lazy">，x=-10{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="X = - 10" style="vertical-align:-6px;height:22px;width:63px" loading="lazy">而且y=17{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="Y = 17" style="vertical-align:-6px;height:22px;width:46px" loading="lazy">．

          (c)
          利用给定的初始条件来确定系统的精确解。
            评估你的分数
            关键概念
            求解耦合微分方程

            问题2 d

            分数:2
            (d)
            通过考虑适当的限制t→∞{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="T向右箭头无穷大" style="vertical-align:-6px;height:22px;width:46px" loading="lazy">，确定变量的长期行为x{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">而且y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">．
              评估你的分数

              问题3

              分数:5

              两个变量的变化率，x{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">而且y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">，可由下列微分方程组描述:

              dxdt=4x+y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分子dx除以分母dt端分数等于4x + y" style="vertical-align:-17px;height:47px;width:91px" loading="lazy">

              dydt=-5x-2y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分子dy除以分母dt端分数等于- 5x - 2y" style="vertical-align:-17px;height:47px;width:117px" loading="lazy">

              矩阵41-5-2{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="开括号表4行1行cell - 5结束cell - 2结束cell结束表闭括号" style="vertical-align:-21px;height:54px;width:75px" loading="lazy">具有特征值3{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="3." style="vertical-align:-6px;height:22px;width:10px" loading="lazy">而且-1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="- 1" style="vertical-align:-6px;height:22px;width:23px" loading="lazy">对应的特征向量1-1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="开括号表第1行单元格- 1结束单元格结束表闭括号" style="vertical-align:-21px;height:54px;width:45px" loading="lazy">而且1-5{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="开括号表第1行单元格- 5结束单元格结束表闭括号" style="vertical-align:-21px;height:54px;width:45px" loading="lazy">．最初x=1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="X = 1" style="vertical-align:-6px;height:22px;width:37px" loading="lazy">而且y=3.{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="Y = 3。" style="vertical-align:-6px;height:22px;width:42px" loading="lazy">

              一)
              利用上述信息求出微分方程组的精确解。
                评估你的分数
                关键概念
                求解耦合微分方程

                问题3 b

                标志:6
                (b)
                使用步长为的欧拉方法0.2{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="0.2" style="vertical-align:-6px;height:22px;width:24px" loading="lazy">求值的近似值x{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">而且y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">当t=1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="T = 1" style="vertical-align:-6px;height:22px;width:33px" loading="lazy">．
                  评估你的分数
                  关键概念
                  欧拉方法:耦合系统
                  误差百分比

                  问题3 c

                  分数:5
                  (c)
                  (我)
                  求部分(b)的近似值与的准确值相比的误差百分比x{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">而且y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">当t=1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="T = 1" style="vertical-align:-6px;height:22px;width:33px" loading="lazy">．

                  (2)
                  评论(b)部分近似的准确性，并解释如何改进。
                    评估你的分数

                    问题4

                    标志:4

                    对于下列耦合微分方程组的每一个通解，

                    (我)
                    画出系统的相位图

                    (2)
                    陈述这一点(0, 0){"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="左括号0，空格0右括号" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">是一个稳定的平衡点还是一个不稳定的平衡点。

                    (一)
                    x=Aet11+Be2t-13{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体x = A e的t次幂开括号表第1行第1行结束表闭括号加上B e的2t次幂结束指数开括号表第1行结束单元格第3行结束表闭括号" style="vertical-align:-21px;height:54px;width:181px" loading="lazy">
                      评估你的分数
                      关键概念
                      阶段的肖像
                      平衡分

                      问题4 b

                      标志:4
                      (b)

                      x=Ae-2t41+Be-3t15{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体x = A e的- 2t次方结束指数的开括号表第4行第1行结束表结束括号加上B e的- 3t次方结束指数的开括号表第1行第5行结束表结束括号" style="vertical-align:-21px;height:54px;width:195px" loading="lazy">

                        评估你的分数
                        关键概念
                        阶段的肖像
                        平衡分

                        问题4摄氏度

                        标志:4
                        (c)
                        x=Aet-11+Be-t43{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体x = A e的t次幂开括号表第一行单元格- 1结束单元格第一行结束表闭括号加上B e的- t次幂结束指数开括号表第4行第三行结束表闭括号" style="vertical-align:-21px;height:54px;width:184px" loading="lazy">
                          评估你的分数
                          关键概念
                          阶段的肖像
                          平衡分

                          问题5

                          标志:4

                          两个变量的行为，x{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">而且y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">，由以下微分方程组建模:

                          dxdt=3x-5y           dydt=x-y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分数分子dx除以分母dt结束分数等于3x - 5y空白分数分子dy除以分母dt结束分数等于x - y" style="vertical-align:-17px;height:47px;width:225px" loading="lazy">

                          在哪里x=1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="X = 1" style="vertical-align:-6px;height:22px;width:37px" loading="lazy">而且y=1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="Y = 1" style="vertical-align:-6px;height:22px;width:37px" loading="lazy">当t=0{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="T = 0" style="vertical-align:-6px;height:22px;width:33px" loading="lazy">．

                          矩阵3-51-1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="开括号表第3行单元格- 5结束单元格第1行单元格- 1结束单元格结束表闭括号" style="vertical-align:-21px;height:54px;width:62px" loading="lazy">具有特征值1+i{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="1加上I" style="vertical-align:-6px;height:22px;width:32px" loading="lazy">而且1-i{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="1减去I" style="vertical-align:-6px;height:22px;width:32px" loading="lazy">．

                          (一)
                          (我)
                          求的值​dxdt{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分子dx除以分母dt结束分数" style="vertical-align:-17px;height:47px;width:29px" loading="lazy">而且​dydt{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分子dy除以分母dt结束分数" style="vertical-align:-17px;height:47px;width:29px" loading="lazy">在这一点上(0, 1){"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="左括号0，空格1右括号" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">．

                          (2)
                          在给定初始条件下，画出系统的相位图。
                            评估你的分数
                            关键概念
                            特征值和特征向量
                            阶段的肖像

                            问题5 b

                            分数:3

                            建议用系统来描述变量可能更好

                            dxdt=-3x-5y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分子dx除以分母dt端分数等于- 3x - 5y" style="vertical-align:-17px;height:47px;width:117px" loading="lazy">dydt=x+y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分子dy除以分母dt端分数等于x + y" style="vertical-align:-17px;height:47px;width:82px" loading="lazy">

                            初始条件相同。

                            (b)计算矩阵的特征值-3-511{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="开括号表行单元格- 3结束单元格单元格- 5结束单元格第1行1结束表闭括号" style="vertical-align:-21px;height:54px;width:75px" loading="lazy">

                              评估你的分数

                              问题5度

                              分数:2
                              (c)
                              因此，请描述(a)(ii)部分的相位图将如何变化以表示这个新的微分方程组。
                                评估你的分数

                                问题6

                                分数:2

                                科学家们一直在追踪水平，x{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">而且y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">，并记录每种污染物的水平相对于历史基线数据(因此，正值表示高于基线值，负值表示低于基线值)。根据已知的污染物之间的相互作用以及与大气中其他物质的相互作用，科学家们建议用以下微分方程系统对这种情况进行建模:

                                dxdt=x-2y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分子dx除以分母dt端分数等于x - 2y" style="vertical-align:-17px;height:47px;width:91px" loading="lazy">

                                dydt=x-y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分子dy除以分母dt端分数等于x - y" style="vertical-align:-17px;height:47px;width:82px" loading="lazy">

                                一)求的值​dxdt{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分子dx除以分母dt结束分数" style="vertical-align:-17px;height:47px;width:29px" loading="lazy">而且dydt{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分子dy除以分母dt，结束分数" style="vertical-align:-17px;height:47px;width:28px" loading="lazy">在这一点上(1, 0){"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="左括号1，空格0右括号" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">而且(0, 1){"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="左括号0，空格1右括号" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">．

                                  评估你的分数
                                  关键概念
                                  特征值和特征向量

                                  问题6 b

                                  分数:3
                                  (b)求出矩阵的特征值1-21-1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="开括号表第一行单元格- 2结束单元格第一行单元格- 1结束单元格结束表闭括号" style="vertical-align:-21px;height:54px;width:62px" loading="lazy">．



                                    评估你的分数

                                    问题6摄氏度

                                    标志:4

                                    在研究开始时，两种污染物都高于基线水平x=5{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="X = 5" style="vertical-align:-6px;height:22px;width:37px" loading="lazy">而且y=3{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="Y = 3" style="vertical-align:-6px;height:22px;width:37px" loading="lazy">．

                                    (c)
                                    使用上述信息来描绘一个相肖像，显示的长期行为x{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">而且y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">．
                                      评估你的分数
                                      关键概念
                                      阶段的肖像

                                      问题7

                                      标志:6

                                      有两种细菌，X{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="X" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">而且Y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="Y" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">它们正在一个研究实验室的培养板上生长。从过去对这两种细菌及其相互作用的研究中，研究人员认为，这两个种群的生长可能由以下微分方程表示

                                      dxdt=-5x+4y           dydt=-8x+7y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分数分子dx除以分母dt结束分数等于- 5x + 4y空白分数分子dy除以分母dt结束分数等于- 8x + 7y" style="vertical-align:-17px;height:47px;width:277px" loading="lazy">

                                      适用于x{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">千和y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">数千种细菌类型X{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="X" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">而且Y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="Y" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">分别。最初，这个板包含20 000{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="000" style="vertical-align:-6px;height:22px;width:48px" loading="lazy">细菌类型X{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="X" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">而且21 000{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="21 000" style="vertical-align:-6px;height:22px;width:48px" loading="lazy">类型的Y.{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="Y。" style="vertical-align:-6px;height:22px;width:20px" loading="lazy">

                                      一)

                                      找出矩阵的特征值和相应的特征向量-54-87{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="开括号表行单元格- 5结束单元格4行单元格- 8结束单元格7结束表闭括号" style="vertical-align:-21px;height:54px;width:62px" loading="lazy">．

                                        评估你的分数
                                        关键概念
                                        特征值和特征向量

                                        问题7 b

                                        分数:2
                                        (b)求的值​dxdt{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分子dx除以分母dt结束分数" style="vertical-align:-17px;height:47px;width:29px" loading="lazy">而且​dydt{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="分子dy除以分母dt结束分数" style="vertical-align:-17px;height:47px;width:29px" loading="lazy">当t=0.{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="T = 0。" style="vertical-align:-6px;height:22px;width:38px" loading="lazy">



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                                          问题7 c

                                          标志:4
                                          (c)
                                          描绘出这两种细菌种群生长的可能轨迹，确保指出任何渐近行为。
                                            评估你的分数
                                            关键概念
                                            勾画溶液轨迹


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