How do I convert this second order differential equation to two first order differential equations?












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The equation is $$m{ddot x} + kx + gsinθ = 0.$$



I know I have to convert it to the form ${dot y}_1 = y_2$ and ${dot y}_2 = text{something}$.



However I am very inexperienced and very confused on how to find $y_1$ and $y_2$ from this initial equation.






calculus linear-algebra integration derivatives







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    0












    $begingroup$


    The equation is $$m{ddot x} + kx + gsinθ = 0.$$



    I know I have to convert it to the form ${dot y}_1 = y_2$ and ${dot y}_2 = text{something}$.



    However I am very inexperienced and very confused on how to find $y_1$ and $y_2$ from this initial equation.






    calculus linear-algebra integration derivatives







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      The equation is $$m{ddot x} + kx + gsinθ = 0.$$



      I know I have to convert it to the form ${dot y}_1 = y_2$ and ${dot y}_2 = text{something}$.



      However I am very inexperienced and very confused on how to find $y_1$ and $y_2$ from this initial equation.






      calculus linear-algebra integration derivatives







      share|cite|improve this question











      $endgroup$




      The equation is $$m{ddot x} + kx + gsinθ = 0.$$



      I know I have to convert it to the form ${dot y}_1 = y_2$ and ${dot y}_2 = text{something}$.



      However I am very inexperienced and very confused on how to find $y_1$ and $y_2$ from this initial equation.






      calculus linear-algebra integration derivatives




      calculus linear-algebra integration derivatives






      share|cite|improve this question















      share|cite|improve this question













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      share|cite|improve this question








      edited Dec 13 '18 at 22:45









      Tianlalu

      3,08421138




      3,08421138










      asked Dec 13 '18 at 22:35









      yeetuscleetusyeetuscleetus

      41




      41






















          1 Answer
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          0












          $begingroup$

          Let's pose $dot{x} = v$ (i.e. velocity). Then:



          $$begin{cases}
          m dot{v} + kx + gsintheta = 0\
          dot{x} = v
          end{cases} Rightarrow begin{cases}
          dot{v} = -frac{k}{m}x - frac{g}{m}sintheta\
          dot{x} = v.
          end{cases}$$



          Maybe, further calculation can be done for $theta$. But if you don't specify its meaning, then these just are meaningless suppositions.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            θ is an arbitrary, constant angle, that represents the angle above horizontal for a slanted plane. I am not experienced enough to know if that will affect the solution to the linear system.
            $endgroup$
            – yeetuscleetus
            Dec 13 '18 at 22:46








          • 1




            $begingroup$
            @erics: The value of the parameter $theta$ certainly affects the solution set: If $sin theta = 0$, the system is homogeneous, and in particular $x(t) = 0$ is a solution (this makes good intuitive sense in terms of the physical system, too). If not, the system is not homogeneous and $x(t) = 0$ is not a solution.
            $endgroup$
            – Travis
            Dec 13 '18 at 22:54










          • $begingroup$
            is there any relationship between $theta$ and $x$? I mean, can I write $theta$ as a function of $x$ (or vice versa)? If not, my answer is complete.
            $endgroup$
            – the_candyman
            Dec 13 '18 at 23:02












          • $begingroup$
            @the_candyman No, there is no relationship. Your answer is correct, thank you!
            $endgroup$
            – yeetuscleetus
            Dec 14 '18 at 0:20











          Your Answer





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          1 Answer
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          oldest

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          1 Answer
          1






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          active

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          0












          $begingroup$

          Let's pose $dot{x} = v$ (i.e. velocity). Then:



          $$begin{cases}
          m dot{v} + kx + gsintheta = 0\
          dot{x} = v
          end{cases} Rightarrow begin{cases}
          dot{v} = -frac{k}{m}x - frac{g}{m}sintheta\
          dot{x} = v.
          end{cases}$$



          Maybe, further calculation can be done for $theta$. But if you don't specify its meaning, then these just are meaningless suppositions.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            θ is an arbitrary, constant angle, that represents the angle above horizontal for a slanted plane. I am not experienced enough to know if that will affect the solution to the linear system.
            $endgroup$
            – yeetuscleetus
            Dec 13 '18 at 22:46








          • 1




            $begingroup$
            @erics: The value of the parameter $theta$ certainly affects the solution set: If $sin theta = 0$, the system is homogeneous, and in particular $x(t) = 0$ is a solution (this makes good intuitive sense in terms of the physical system, too). If not, the system is not homogeneous and $x(t) = 0$ is not a solution.
            $endgroup$
            – Travis
            Dec 13 '18 at 22:54










          • $begingroup$
            is there any relationship between $theta$ and $x$? I mean, can I write $theta$ as a function of $x$ (or vice versa)? If not, my answer is complete.
            $endgroup$
            – the_candyman
            Dec 13 '18 at 23:02












          • $begingroup$
            @the_candyman No, there is no relationship. Your answer is correct, thank you!
            $endgroup$
            – yeetuscleetus
            Dec 14 '18 at 0:20
















          0












          $begingroup$

          Let's pose $dot{x} = v$ (i.e. velocity). Then:



          $$begin{cases}
          m dot{v} + kx + gsintheta = 0\
          dot{x} = v
          end{cases} Rightarrow begin{cases}
          dot{v} = -frac{k}{m}x - frac{g}{m}sintheta\
          dot{x} = v.
          end{cases}$$



          Maybe, further calculation can be done for $theta$. But if you don't specify its meaning, then these just are meaningless suppositions.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            θ is an arbitrary, constant angle, that represents the angle above horizontal for a slanted plane. I am not experienced enough to know if that will affect the solution to the linear system.
            $endgroup$
            – yeetuscleetus
            Dec 13 '18 at 22:46








          • 1




            $begingroup$
            @erics: The value of the parameter $theta$ certainly affects the solution set: If $sin theta = 0$, the system is homogeneous, and in particular $x(t) = 0$ is a solution (this makes good intuitive sense in terms of the physical system, too). If not, the system is not homogeneous and $x(t) = 0$ is not a solution.
            $endgroup$
            – Travis
            Dec 13 '18 at 22:54










          • $begingroup$
            is there any relationship between $theta$ and $x$? I mean, can I write $theta$ as a function of $x$ (or vice versa)? If not, my answer is complete.
            $endgroup$
            – the_candyman
            Dec 13 '18 at 23:02












          • $begingroup$
            @the_candyman No, there is no relationship. Your answer is correct, thank you!
            $endgroup$
            – yeetuscleetus
            Dec 14 '18 at 0:20














          0












          0








          0





          $begingroup$

          Let's pose $dot{x} = v$ (i.e. velocity). Then:



          $$begin{cases}
          m dot{v} + kx + gsintheta = 0\
          dot{x} = v
          end{cases} Rightarrow begin{cases}
          dot{v} = -frac{k}{m}x - frac{g}{m}sintheta\
          dot{x} = v.
          end{cases}$$



          Maybe, further calculation can be done for $theta$. But if you don't specify its meaning, then these just are meaningless suppositions.






          share|cite|improve this answer











          $endgroup$



          Let's pose $dot{x} = v$ (i.e. velocity). Then:



          $$begin{cases}
          m dot{v} + kx + gsintheta = 0\
          dot{x} = v
          end{cases} Rightarrow begin{cases}
          dot{v} = -frac{k}{m}x - frac{g}{m}sintheta\
          dot{x} = v.
          end{cases}$$



          Maybe, further calculation can be done for $theta$. But if you don't specify its meaning, then these just are meaningless suppositions.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 13 '18 at 23:04

























          answered Dec 13 '18 at 22:38









          the_candymanthe_candyman

          8,97832145




          8,97832145












          • $begingroup$
            θ is an arbitrary, constant angle, that represents the angle above horizontal for a slanted plane. I am not experienced enough to know if that will affect the solution to the linear system.
            $endgroup$
            – yeetuscleetus
            Dec 13 '18 at 22:46








          • 1




            $begingroup$
            @erics: The value of the parameter $theta$ certainly affects the solution set: If $sin theta = 0$, the system is homogeneous, and in particular $x(t) = 0$ is a solution (this makes good intuitive sense in terms of the physical system, too). If not, the system is not homogeneous and $x(t) = 0$ is not a solution.
            $endgroup$
            – Travis
            Dec 13 '18 at 22:54










          • $begingroup$
            is there any relationship between $theta$ and $x$? I mean, can I write $theta$ as a function of $x$ (or vice versa)? If not, my answer is complete.
            $endgroup$
            – the_candyman
            Dec 13 '18 at 23:02












          • $begingroup$
            @the_candyman No, there is no relationship. Your answer is correct, thank you!
            $endgroup$
            – yeetuscleetus
            Dec 14 '18 at 0:20


















          • $begingroup$
            θ is an arbitrary, constant angle, that represents the angle above horizontal for a slanted plane. I am not experienced enough to know if that will affect the solution to the linear system.
            $endgroup$
            – yeetuscleetus
            Dec 13 '18 at 22:46








          • 1




            $begingroup$
            @erics: The value of the parameter $theta$ certainly affects the solution set: If $sin theta = 0$, the system is homogeneous, and in particular $x(t) = 0$ is a solution (this makes good intuitive sense in terms of the physical system, too). If not, the system is not homogeneous and $x(t) = 0$ is not a solution.
            $endgroup$
            – Travis
            Dec 13 '18 at 22:54










          • $begingroup$
            is there any relationship between $theta$ and $x$? I mean, can I write $theta$ as a function of $x$ (or vice versa)? If not, my answer is complete.
            $endgroup$
            – the_candyman
            Dec 13 '18 at 23:02












          • $begingroup$
            @the_candyman No, there is no relationship. Your answer is correct, thank you!
            $endgroup$
            – yeetuscleetus
            Dec 14 '18 at 0:20
















          $begingroup$
          θ is an arbitrary, constant angle, that represents the angle above horizontal for a slanted plane. I am not experienced enough to know if that will affect the solution to the linear system.
          $endgroup$
          – yeetuscleetus
          Dec 13 '18 at 22:46






          $begingroup$
          θ is an arbitrary, constant angle, that represents the angle above horizontal for a slanted plane. I am not experienced enough to know if that will affect the solution to the linear system.
          $endgroup$
          – yeetuscleetus
          Dec 13 '18 at 22:46






          1




          1




          $begingroup$
          @erics: The value of the parameter $theta$ certainly affects the solution set: If $sin theta = 0$, the system is homogeneous, and in particular $x(t) = 0$ is a solution (this makes good intuitive sense in terms of the physical system, too). If not, the system is not homogeneous and $x(t) = 0$ is not a solution.
          $endgroup$
          – Travis
          Dec 13 '18 at 22:54




          $begingroup$
          @erics: The value of the parameter $theta$ certainly affects the solution set: If $sin theta = 0$, the system is homogeneous, and in particular $x(t) = 0$ is a solution (this makes good intuitive sense in terms of the physical system, too). If not, the system is not homogeneous and $x(t) = 0$ is not a solution.
          $endgroup$
          – Travis
          Dec 13 '18 at 22:54












          $begingroup$
          is there any relationship between $theta$ and $x$? I mean, can I write $theta$ as a function of $x$ (or vice versa)? If not, my answer is complete.
          $endgroup$
          – the_candyman
          Dec 13 '18 at 23:02






          $begingroup$
          is there any relationship between $theta$ and $x$? I mean, can I write $theta$ as a function of $x$ (or vice versa)? If not, my answer is complete.
          $endgroup$
          – the_candyman
          Dec 13 '18 at 23:02














          $begingroup$
          @the_candyman No, there is no relationship. Your answer is correct, thank you!
          $endgroup$
          – yeetuscleetus
          Dec 14 '18 at 0:20




          $begingroup$
          @the_candyman No, there is no relationship. Your answer is correct, thank you!
          $endgroup$
          – yeetuscleetus
          Dec 14 '18 at 0:20


















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