An Approach to Mathematic Functions Basics

by Frank C. Fung ( 1st published in December, 2014. )

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Section XXVIII - The Differential Equation involving [ R ] and [ Theta ]

Summary for the Section :

• In this Section , we shall go thru the derivation process for setting up :
  • the Differential Equation involving [ R ] and [ Theta ] .

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Recalling the Differential Equation from the last Section :

• Let us bring-in again this Differential Equation from the last Section :
  • 

    { the [ core Differential Equation governing the behavior of the Satellite ] }

  And we shall now re-write this equation in this format :

  • 

    for our usage below .

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The derivatives of [ Theta ] with respect to time :

• Since we have already established in Section XXVI that :
  • 

  we can now express the { first-derivative of [ Theta ] with respect to time } as :

  • 

• Taking the derivative thereof with respect to time then yields us this relation :
  • 

    yielding :

  • 

  Consequently :

  • 

• Let us bring back , at this point , this relation from above :
  • 

  Substituting this into the equation immediately above then yields us this relation :

  • 

  And consequently , we have this final equation :

  • 

    for the { second-derivative of [ Theta ] with respect to time } .

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The Derivative of [ R ] with respect to time :

• Let us now bring-in the { first derivative of [ R ] with respect to time } in this format :
  • 

  Taking the derivative thereof with respect to time then yields us this relation :

  • 

    yielding :

  • 

  Therefore :

  • 

• Let us now bring-in these 2 relations established above :
  • 

    and

  • 

  Substituting these into the equation immediately before then yields us this relation :

  • 

  And consequently , we have this final equation :

  • 

    for the { second-derivative of [ R ] with respect to time } .

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Setting up the { Differential Equation involving [ R ] and [ Theta ] } :

Let us bring-in again the original Differential Equation from the beginning of this Section :
• i.e. this equation :
  • 

Let us recall that we have already established above :

• 

  and

• 

Substituting these into the equation immediately before then yields us this relation :

• 

  yielding :

• 

This is an important equation that we shall be coming back to in Section XXXIV of this paper ,

• and we shall now label this equation as :
  • the { Differential Equation involvling [ R ] and [ Theta ] in its First Format } .

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The { Differential Equation involving [ R ] and [ Theta ] in its Second Format } :

• Let us bring-in this Differential Equation immediately above :
  • 

  Let us now multiply throughout by :

  • [ R-square ] over [ K-square ] .

  We then arrive at this final equation below :

  • 

• And we shall now label this Differential Equation as :
  • the { Differential Equation involving [ R ] and [ Theta ] in its Second Format }
    • for our further study in Section XXXII of this paper .

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go to the next section : Section XXIX - Setting up the Standardized Set of Initial Conditions

go to the last section : Section XXVII - Setting up the Central Force influencing the Satellite

return to the MATH Functions Basics HomePage

Original dated 2014-12-07