TDCC Laboratory

   HOME JAPANESE                                         

A Solution Method (?) of the 4th Order Equation


1. Preface

     Apart from the ordinary method, a solution method of the 4th order equation which we have thought of is introduced.      In this method, we lead the solution of the 4th order equation into the solutions of the 2nd order twin equations while the coefficient of the each term of these twin equations is determined by solving the 3rd order equation.      In another words, this method is the way to replace the solution of the 4th order equation into the solutions of a set of the 2nd order twin equations and the 3rd order equation.
     By the way, a solution method of the 5th order equation does not exist.

2. Guideline of the Proposed Method

     We consider the 4th order equation given as the following;
           .             (1)
     We intend to rewrite the above equation into the followings;
           ,               (2)
           ,                   (3)
           .               (4)
     Eq.(2) indicates the 2nd order equation concerning f(x), and f(x) itself is the 2nd order equation given by Eq.(3).      Eq.(4) is settled as the condition of the discriminant of Eq.(2) becomes as the following;
           .               (5)
     According to the above condition, the solution f(x) of Eq.(2) is expressed by the rational equation written as
           .               (6)
     If we substitute Eq.(3) into the above equation, the 2nd order twin equations (in the same compound order) yields as given in the following;
           .               (7)
Thus, these 2nd order twin equations should be solved instead of the 4th order equation.
     The remained problem is to decide the coefficients a, b, c, d and e.      The calculations for the determination of these coefficients are described in the next section.

3. Obtaining of the Coefficients a, b, c, d and e

     If we substitute Eqs.(3) and (4) into Eq.(2), we obtain that
           .               (8)
     Comparing the above equation to the original one of Eq.(1), it is obtained that
           ,                                             (9)
           ,                (10)
           ,                       (11)
           .                   (12)

     From Eq.(9),
           .                                           (13)
     If we substitute Eq.(9) into Eqs.(10), (11) and (12), we obtain that
           ,                 (14)
           ,                        (15)
           ,                            (16)

     From Eq.(15),
           .                        (17)
     If we take the square of the each side of the above equation, we may write
           .                    (18)

     If we substitute Eqs.(14) and (16) into the above equation, the 3rd order equation for obtaining the coefficient b+c as in the following;
           ,                    (19)
           ,                                           (20)
           ,                                 (21)
           .                  (22)

From Eq.(14),
           .                  (23)
From Eq.(16),
           .                       (24)

4, Determination of the Coefficients a, b+c, d and e

     Coefficient a is obtained by Eq.(13).      Compound coefficients b+c is obtained by the real number's solution of Eq.(19).      If Eq.(19) has three solutions of real number, we may choose arbitrary one of three ones.      With the use of this solution of b+c, d and e are obtained by Eqs.(23) and (24) respectively.

5. Conclusion

     Substituting the coefficients obtained in the previous section into Eq.(7), the 2nd order twin equations are decided.      By solving thus decided 2nd order twin equations, four solutions of the 4th order equation of Eq.(1) may be obtained.

     [Notation] Whether the solution method mentioned above is truely effective or not is under consideration now.

_ returning to the former page with 'return' of browser _


_ returning to the home page with the following button _


Updated: 2008.11.07, edited by N. Yamamoto.
Revised on Mar. 16, 2015, May 05, 2020, Jan. 31, 2021, May 08, 2021 and Mar. 26, 2022.