Decouple differential equationsI have a system of two Second Order differential equations r 2 r ¨ − r 3 ( φ ˙ 2 + ω 2 ) = − G M r φ ¨ + 2 r ˙ ( φ ˙ + ω ) = 0which I am supposed to decouple using the conservation size ( φ ˙ + ω ) r 2 I have shown, that it is indeed a conservation size, as its derivation is r-times the second equation and therefore zero. However I don't know how this is supposed to help me decoupling the two equations.

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Decouple differential equationsI have a system of two Second Order differential equations      r    2              r      ¨        −      r    3    (                    φ        ˙              2    +      ω    2    )  =  −  G  M  r            φ      ¨        +  2            r      ˙        (            φ      ˙        +  ω  )  =  0which I am supposed to decouple using the conservation size   (            φ      ˙        +  ω  )      r    2   I have shown, that it is indeed a conservation size, as its derivation is r-times the second equation and therefore zero. However I don&#039;t know how this is supposed to help me decoupling the two equations.

Willie Smith · Accepted Answer

Step 1  r      φ    ″    +  2      r    ′    (      φ    ′    +  ω  )  =  0    →              φ      ″                      φ        ′            +      ω        =  −  2            r      ′        r    ln  ⁡      |        φ    ′    +  ω      |    =  −  2  ln  ⁡  [  r      |    +  c      φ    ′    =      C          r      2        −  ω      r    2        r    ′    −      r    3    (  (      φ    ′        )    2    +      ω    2    )  =  −  G  M  =      r    2        r    ′    −      r    3              (              C                  r          2                    −      ω      )        2    −      r    3        ω    2        r    ′    =  r            (              C                  r          2                    −      ω      )        2    +  r      ω    2    −            G      M              r      2                  d      r              d      t        =      r    ′    =            C      2              r      3        −            G      M              r      2        −            2      C      ω        r    +  2  r      ω    2    t  =  ∫            d      r                                C          2                          r          3                    −                        G          M                          r          2                    −                        2          C          ω                r            +      2      r              ω        2              +      c    1  Step 2This integral can be analytically solved. This would be a arduous task, involving a quartic polynomial equation to be solved and then separating the polynomial fraction into four elementary fractions which are each integrable. Too long to be done here.            d      φ              d      t        =      C          r      2        −  ω    →              d      φ              d      r        =  (      C          r      2        −  ω  )            d      r              d      t                  d      φ              d      r        =  (      C          r      2        −  ω  )      (                  C        2                    r        3              −                  G        M                    r        2              −                  2        C        ω            r        +    2    r          ω      2        )    φ  (  r  )  =  ∫  (      C          r      2        −  ω  )      (                  C        2                    r        3              −                  G        M                    r        2              −                  2        C        ω            r        +    2    r          ω      2        )    d  rThis is easy to integrate (expand and integrate each elementary term).So, the problem is solved on parametric form :- The explicit function φ(r)- The implicit function t(r).There is probably no closed form for the inverse function r(t).

Decouple differential equations. I have a system of two Second Order differential equations r^2 ddot{r} - r^3(\dot{\varphi}^2 +\omega^2) =-GM

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