How to change the system in four first order equations?












0












$begingroup$


Consider the coupled spring-mass system with a frictionless table, two masses $m_1$ and $m_2$, and three springs with spring constants $k_1, k_2$, and $k_3$ respectively. The equation of motion for the system are given by:



$m_1frac{d^2x_1}{dt^2}=-(k_1+k_2)x_1+k_2x_2$



$m_2frac{d^2x_2}{dt^2}=k_2x_1-(k_2+k_3)x_2$
enter image description here



Assume that the masses are $m_1 = 2$, $m_2 = frac{9}{4}$, and the spring constants are $k_1=1,k_2=3,k_3=frac{15}{4}$.



I substitute all value in the system above, and able to get



$frac{d^2x_1}{dt^2}=-2x_1+frac{3}{2}x_2$



$frac{d^2x_2}{dt^2}=frac{4}{3}x_1-3x_2$



Then how to convert the system about into four first order equations?










share|cite|improve this question









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    0












    $begingroup$


    Consider the coupled spring-mass system with a frictionless table, two masses $m_1$ and $m_2$, and three springs with spring constants $k_1, k_2$, and $k_3$ respectively. The equation of motion for the system are given by:



    $m_1frac{d^2x_1}{dt^2}=-(k_1+k_2)x_1+k_2x_2$



    $m_2frac{d^2x_2}{dt^2}=k_2x_1-(k_2+k_3)x_2$
    enter image description here



    Assume that the masses are $m_1 = 2$, $m_2 = frac{9}{4}$, and the spring constants are $k_1=1,k_2=3,k_3=frac{15}{4}$.



    I substitute all value in the system above, and able to get



    $frac{d^2x_1}{dt^2}=-2x_1+frac{3}{2}x_2$



    $frac{d^2x_2}{dt^2}=frac{4}{3}x_1-3x_2$



    Then how to convert the system about into four first order equations?










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Consider the coupled spring-mass system with a frictionless table, two masses $m_1$ and $m_2$, and three springs with spring constants $k_1, k_2$, and $k_3$ respectively. The equation of motion for the system are given by:



      $m_1frac{d^2x_1}{dt^2}=-(k_1+k_2)x_1+k_2x_2$



      $m_2frac{d^2x_2}{dt^2}=k_2x_1-(k_2+k_3)x_2$
      enter image description here



      Assume that the masses are $m_1 = 2$, $m_2 = frac{9}{4}$, and the spring constants are $k_1=1,k_2=3,k_3=frac{15}{4}$.



      I substitute all value in the system above, and able to get



      $frac{d^2x_1}{dt^2}=-2x_1+frac{3}{2}x_2$



      $frac{d^2x_2}{dt^2}=frac{4}{3}x_1-3x_2$



      Then how to convert the system about into four first order equations?










      share|cite|improve this question









      $endgroup$




      Consider the coupled spring-mass system with a frictionless table, two masses $m_1$ and $m_2$, and three springs with spring constants $k_1, k_2$, and $k_3$ respectively. The equation of motion for the system are given by:



      $m_1frac{d^2x_1}{dt^2}=-(k_1+k_2)x_1+k_2x_2$



      $m_2frac{d^2x_2}{dt^2}=k_2x_1-(k_2+k_3)x_2$
      enter image description here



      Assume that the masses are $m_1 = 2$, $m_2 = frac{9}{4}$, and the spring constants are $k_1=1,k_2=3,k_3=frac{15}{4}$.



      I substitute all value in the system above, and able to get



      $frac{d^2x_1}{dt^2}=-2x_1+frac{3}{2}x_2$



      $frac{d^2x_2}{dt^2}=frac{4}{3}x_1-3x_2$



      Then how to convert the system about into four first order equations?







      ordinary-differential-equations






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 5 '18 at 10:45









      LTYLTY

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          $begingroup$

          To convert it into a system of four first-order ODEs, you simply define the new variables $$x_3 = frac{dx_1}{dt} qquad x_4 = frac{dx_2}{dt}$$



          The system then becomes



          $$frac{dx_1}{dt} = x_3 qquad frac{dx_2}{dt} = x_4 qquad frac{dx_3}{dt}=-2x_1+frac{3}{2}x_2 qquad frac{dx_4}{dt}=frac{4}{3}x_1-3x_2$$





          However, I am not sure this actually helps you to solve the system. Instead, you might consider using



          begin{align}
          frac{d^2x_1}{dt^2} & =-2x_1+frac{3}{2}x_2 tag{1}\
          frac{d^2x_2}{dt^2} & =frac{4}{3}x_1-3x_2 tag{2}
          end{align}



          Do $frac{d^2}{dt^2}$ of equation $(1)$:



          $$frac{d^4x_1}{dt^4} = -2frac{d^2x_1}{dt^2}+frac 32 frac{d^2x_2}{dt^2}$$



          Plug in the expression for $frac{d^2x_2}{dt^2} $ from equation $(2)$:



          $$frac{d^4x_1}{dt^4} = -2frac{d^2x_1}{dt^2}+frac 32 bigg(frac{4}{3}x_1-3x_2bigg) = -2frac{d^2x_1}{dt^2} + 2x_1-frac 92 x_2$$



          Then use equation $(1)$ to plug in expression for $x_2$:



          $$frac{d^4x_1}{dt^4} = -2frac{d^2x_1}{dt^2} + 2x_1-frac 92 bigg(frac 23 frac{d^2x_1}{dt^2} + frac 43 x_1 bigg) = -5frac{d^2x_1}{dt^2} -4x_1$$



          $$implies frac{d^4x_1}{dt^4}+5frac{d^2x_1}{dt^2} + 4x_1=0$$



          This is now a homogeneous ODE with constant coefficients in $x_1$, which can easily be solved, giving



          $$x_1(t) = Acos (2t)+Bsin (2t) + Ccos (t) + Dsin (t)$$



          Plugging this back into equation $(1)$, we find



          $$x_2(t) = -frac 43 Acos (2t) -frac 43 Bsin (2t)+ frac 23 Ccos (t) + frac 23 Dsin (t)$$



          Finally, use the initial conditions (you should have four of them) to find the constants $A,B,C,D$.






          share|cite|improve this answer











          $endgroup$





















            0












            $begingroup$

            The matrix $A=pmatrix{-2&frac32\frac43&-3}$ on the right side of the system has the characteristic polynomial
            $$
            0=(q+2)(q+3)-2=q^2+5q+4=(q+4)(q+1).
            $$

            For the eigenvalue $q=-1$ a left eigenvector is $(4,3)$ so that
            $$
            (4x_1+3x_2)''=-(4x_1+3x_2)implies 4x_1+3x_2=a_1cos(t)+b_1sin(t)
            $$

            For the eigenvalue $q=-4$ a left eigenvector is $(2,-3)$ so that
            $$
            (2x_1-3x_2)''=-4(2x_1-3x_2)implies 2x_1-3x_2=a_2cos(2t)+b_2sin(2)
            $$

            Now the solutions $x_1$, $x_2$ can be recombined and used to compare against the numerical solution of the first order system as per the other answer.






            share|cite|improve this answer









            $endgroup$













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              $begingroup$

              To convert it into a system of four first-order ODEs, you simply define the new variables $$x_3 = frac{dx_1}{dt} qquad x_4 = frac{dx_2}{dt}$$



              The system then becomes



              $$frac{dx_1}{dt} = x_3 qquad frac{dx_2}{dt} = x_4 qquad frac{dx_3}{dt}=-2x_1+frac{3}{2}x_2 qquad frac{dx_4}{dt}=frac{4}{3}x_1-3x_2$$





              However, I am not sure this actually helps you to solve the system. Instead, you might consider using



              begin{align}
              frac{d^2x_1}{dt^2} & =-2x_1+frac{3}{2}x_2 tag{1}\
              frac{d^2x_2}{dt^2} & =frac{4}{3}x_1-3x_2 tag{2}
              end{align}



              Do $frac{d^2}{dt^2}$ of equation $(1)$:



              $$frac{d^4x_1}{dt^4} = -2frac{d^2x_1}{dt^2}+frac 32 frac{d^2x_2}{dt^2}$$



              Plug in the expression for $frac{d^2x_2}{dt^2} $ from equation $(2)$:



              $$frac{d^4x_1}{dt^4} = -2frac{d^2x_1}{dt^2}+frac 32 bigg(frac{4}{3}x_1-3x_2bigg) = -2frac{d^2x_1}{dt^2} + 2x_1-frac 92 x_2$$



              Then use equation $(1)$ to plug in expression for $x_2$:



              $$frac{d^4x_1}{dt^4} = -2frac{d^2x_1}{dt^2} + 2x_1-frac 92 bigg(frac 23 frac{d^2x_1}{dt^2} + frac 43 x_1 bigg) = -5frac{d^2x_1}{dt^2} -4x_1$$



              $$implies frac{d^4x_1}{dt^4}+5frac{d^2x_1}{dt^2} + 4x_1=0$$



              This is now a homogeneous ODE with constant coefficients in $x_1$, which can easily be solved, giving



              $$x_1(t) = Acos (2t)+Bsin (2t) + Ccos (t) + Dsin (t)$$



              Plugging this back into equation $(1)$, we find



              $$x_2(t) = -frac 43 Acos (2t) -frac 43 Bsin (2t)+ frac 23 Ccos (t) + frac 23 Dsin (t)$$



              Finally, use the initial conditions (you should have four of them) to find the constants $A,B,C,D$.






              share|cite|improve this answer











              $endgroup$


















                2












                $begingroup$

                To convert it into a system of four first-order ODEs, you simply define the new variables $$x_3 = frac{dx_1}{dt} qquad x_4 = frac{dx_2}{dt}$$



                The system then becomes



                $$frac{dx_1}{dt} = x_3 qquad frac{dx_2}{dt} = x_4 qquad frac{dx_3}{dt}=-2x_1+frac{3}{2}x_2 qquad frac{dx_4}{dt}=frac{4}{3}x_1-3x_2$$





                However, I am not sure this actually helps you to solve the system. Instead, you might consider using



                begin{align}
                frac{d^2x_1}{dt^2} & =-2x_1+frac{3}{2}x_2 tag{1}\
                frac{d^2x_2}{dt^2} & =frac{4}{3}x_1-3x_2 tag{2}
                end{align}



                Do $frac{d^2}{dt^2}$ of equation $(1)$:



                $$frac{d^4x_1}{dt^4} = -2frac{d^2x_1}{dt^2}+frac 32 frac{d^2x_2}{dt^2}$$



                Plug in the expression for $frac{d^2x_2}{dt^2} $ from equation $(2)$:



                $$frac{d^4x_1}{dt^4} = -2frac{d^2x_1}{dt^2}+frac 32 bigg(frac{4}{3}x_1-3x_2bigg) = -2frac{d^2x_1}{dt^2} + 2x_1-frac 92 x_2$$



                Then use equation $(1)$ to plug in expression for $x_2$:



                $$frac{d^4x_1}{dt^4} = -2frac{d^2x_1}{dt^2} + 2x_1-frac 92 bigg(frac 23 frac{d^2x_1}{dt^2} + frac 43 x_1 bigg) = -5frac{d^2x_1}{dt^2} -4x_1$$



                $$implies frac{d^4x_1}{dt^4}+5frac{d^2x_1}{dt^2} + 4x_1=0$$



                This is now a homogeneous ODE with constant coefficients in $x_1$, which can easily be solved, giving



                $$x_1(t) = Acos (2t)+Bsin (2t) + Ccos (t) + Dsin (t)$$



                Plugging this back into equation $(1)$, we find



                $$x_2(t) = -frac 43 Acos (2t) -frac 43 Bsin (2t)+ frac 23 Ccos (t) + frac 23 Dsin (t)$$



                Finally, use the initial conditions (you should have four of them) to find the constants $A,B,C,D$.






                share|cite|improve this answer











                $endgroup$
















                  2












                  2








                  2





                  $begingroup$

                  To convert it into a system of four first-order ODEs, you simply define the new variables $$x_3 = frac{dx_1}{dt} qquad x_4 = frac{dx_2}{dt}$$



                  The system then becomes



                  $$frac{dx_1}{dt} = x_3 qquad frac{dx_2}{dt} = x_4 qquad frac{dx_3}{dt}=-2x_1+frac{3}{2}x_2 qquad frac{dx_4}{dt}=frac{4}{3}x_1-3x_2$$





                  However, I am not sure this actually helps you to solve the system. Instead, you might consider using



                  begin{align}
                  frac{d^2x_1}{dt^2} & =-2x_1+frac{3}{2}x_2 tag{1}\
                  frac{d^2x_2}{dt^2} & =frac{4}{3}x_1-3x_2 tag{2}
                  end{align}



                  Do $frac{d^2}{dt^2}$ of equation $(1)$:



                  $$frac{d^4x_1}{dt^4} = -2frac{d^2x_1}{dt^2}+frac 32 frac{d^2x_2}{dt^2}$$



                  Plug in the expression for $frac{d^2x_2}{dt^2} $ from equation $(2)$:



                  $$frac{d^4x_1}{dt^4} = -2frac{d^2x_1}{dt^2}+frac 32 bigg(frac{4}{3}x_1-3x_2bigg) = -2frac{d^2x_1}{dt^2} + 2x_1-frac 92 x_2$$



                  Then use equation $(1)$ to plug in expression for $x_2$:



                  $$frac{d^4x_1}{dt^4} = -2frac{d^2x_1}{dt^2} + 2x_1-frac 92 bigg(frac 23 frac{d^2x_1}{dt^2} + frac 43 x_1 bigg) = -5frac{d^2x_1}{dt^2} -4x_1$$



                  $$implies frac{d^4x_1}{dt^4}+5frac{d^2x_1}{dt^2} + 4x_1=0$$



                  This is now a homogeneous ODE with constant coefficients in $x_1$, which can easily be solved, giving



                  $$x_1(t) = Acos (2t)+Bsin (2t) + Ccos (t) + Dsin (t)$$



                  Plugging this back into equation $(1)$, we find



                  $$x_2(t) = -frac 43 Acos (2t) -frac 43 Bsin (2t)+ frac 23 Ccos (t) + frac 23 Dsin (t)$$



                  Finally, use the initial conditions (you should have four of them) to find the constants $A,B,C,D$.






                  share|cite|improve this answer











                  $endgroup$



                  To convert it into a system of four first-order ODEs, you simply define the new variables $$x_3 = frac{dx_1}{dt} qquad x_4 = frac{dx_2}{dt}$$



                  The system then becomes



                  $$frac{dx_1}{dt} = x_3 qquad frac{dx_2}{dt} = x_4 qquad frac{dx_3}{dt}=-2x_1+frac{3}{2}x_2 qquad frac{dx_4}{dt}=frac{4}{3}x_1-3x_2$$





                  However, I am not sure this actually helps you to solve the system. Instead, you might consider using



                  begin{align}
                  frac{d^2x_1}{dt^2} & =-2x_1+frac{3}{2}x_2 tag{1}\
                  frac{d^2x_2}{dt^2} & =frac{4}{3}x_1-3x_2 tag{2}
                  end{align}



                  Do $frac{d^2}{dt^2}$ of equation $(1)$:



                  $$frac{d^4x_1}{dt^4} = -2frac{d^2x_1}{dt^2}+frac 32 frac{d^2x_2}{dt^2}$$



                  Plug in the expression for $frac{d^2x_2}{dt^2} $ from equation $(2)$:



                  $$frac{d^4x_1}{dt^4} = -2frac{d^2x_1}{dt^2}+frac 32 bigg(frac{4}{3}x_1-3x_2bigg) = -2frac{d^2x_1}{dt^2} + 2x_1-frac 92 x_2$$



                  Then use equation $(1)$ to plug in expression for $x_2$:



                  $$frac{d^4x_1}{dt^4} = -2frac{d^2x_1}{dt^2} + 2x_1-frac 92 bigg(frac 23 frac{d^2x_1}{dt^2} + frac 43 x_1 bigg) = -5frac{d^2x_1}{dt^2} -4x_1$$



                  $$implies frac{d^4x_1}{dt^4}+5frac{d^2x_1}{dt^2} + 4x_1=0$$



                  This is now a homogeneous ODE with constant coefficients in $x_1$, which can easily be solved, giving



                  $$x_1(t) = Acos (2t)+Bsin (2t) + Ccos (t) + Dsin (t)$$



                  Plugging this back into equation $(1)$, we find



                  $$x_2(t) = -frac 43 Acos (2t) -frac 43 Bsin (2t)+ frac 23 Ccos (t) + frac 23 Dsin (t)$$



                  Finally, use the initial conditions (you should have four of them) to find the constants $A,B,C,D$.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Dec 5 '18 at 11:00

























                  answered Dec 5 '18 at 10:49









                  glowstonetreesglowstonetrees

                  2,375418




                  2,375418























                      0












                      $begingroup$

                      The matrix $A=pmatrix{-2&frac32\frac43&-3}$ on the right side of the system has the characteristic polynomial
                      $$
                      0=(q+2)(q+3)-2=q^2+5q+4=(q+4)(q+1).
                      $$

                      For the eigenvalue $q=-1$ a left eigenvector is $(4,3)$ so that
                      $$
                      (4x_1+3x_2)''=-(4x_1+3x_2)implies 4x_1+3x_2=a_1cos(t)+b_1sin(t)
                      $$

                      For the eigenvalue $q=-4$ a left eigenvector is $(2,-3)$ so that
                      $$
                      (2x_1-3x_2)''=-4(2x_1-3x_2)implies 2x_1-3x_2=a_2cos(2t)+b_2sin(2)
                      $$

                      Now the solutions $x_1$, $x_2$ can be recombined and used to compare against the numerical solution of the first order system as per the other answer.






                      share|cite|improve this answer









                      $endgroup$


















                        0












                        $begingroup$

                        The matrix $A=pmatrix{-2&frac32\frac43&-3}$ on the right side of the system has the characteristic polynomial
                        $$
                        0=(q+2)(q+3)-2=q^2+5q+4=(q+4)(q+1).
                        $$

                        For the eigenvalue $q=-1$ a left eigenvector is $(4,3)$ so that
                        $$
                        (4x_1+3x_2)''=-(4x_1+3x_2)implies 4x_1+3x_2=a_1cos(t)+b_1sin(t)
                        $$

                        For the eigenvalue $q=-4$ a left eigenvector is $(2,-3)$ so that
                        $$
                        (2x_1-3x_2)''=-4(2x_1-3x_2)implies 2x_1-3x_2=a_2cos(2t)+b_2sin(2)
                        $$

                        Now the solutions $x_1$, $x_2$ can be recombined and used to compare against the numerical solution of the first order system as per the other answer.






                        share|cite|improve this answer









                        $endgroup$
















                          0












                          0








                          0





                          $begingroup$

                          The matrix $A=pmatrix{-2&frac32\frac43&-3}$ on the right side of the system has the characteristic polynomial
                          $$
                          0=(q+2)(q+3)-2=q^2+5q+4=(q+4)(q+1).
                          $$

                          For the eigenvalue $q=-1$ a left eigenvector is $(4,3)$ so that
                          $$
                          (4x_1+3x_2)''=-(4x_1+3x_2)implies 4x_1+3x_2=a_1cos(t)+b_1sin(t)
                          $$

                          For the eigenvalue $q=-4$ a left eigenvector is $(2,-3)$ so that
                          $$
                          (2x_1-3x_2)''=-4(2x_1-3x_2)implies 2x_1-3x_2=a_2cos(2t)+b_2sin(2)
                          $$

                          Now the solutions $x_1$, $x_2$ can be recombined and used to compare against the numerical solution of the first order system as per the other answer.






                          share|cite|improve this answer









                          $endgroup$



                          The matrix $A=pmatrix{-2&frac32\frac43&-3}$ on the right side of the system has the characteristic polynomial
                          $$
                          0=(q+2)(q+3)-2=q^2+5q+4=(q+4)(q+1).
                          $$

                          For the eigenvalue $q=-1$ a left eigenvector is $(4,3)$ so that
                          $$
                          (4x_1+3x_2)''=-(4x_1+3x_2)implies 4x_1+3x_2=a_1cos(t)+b_1sin(t)
                          $$

                          For the eigenvalue $q=-4$ a left eigenvector is $(2,-3)$ so that
                          $$
                          (2x_1-3x_2)''=-4(2x_1-3x_2)implies 2x_1-3x_2=a_2cos(2t)+b_2sin(2)
                          $$

                          Now the solutions $x_1$, $x_2$ can be recombined and used to compare against the numerical solution of the first order system as per the other answer.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Dec 5 '18 at 11:03









                          LutzLLutzL

                          59.2k42057




                          59.2k42057






























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