solve the below equation with respect to x 0.6 ⋅ exp ⁡ ( − 40 x ) + 0.4 ⋅ exp ⁡ ( 10 x ) = 1

Question

solve the below equation with respect to x  0.6  ⋅  exp  ⁡  (            −      40        x    )  +  0.4  ⋅  exp  ⁡  (      10    x    )  =  1

Amadek6 · Accepted Answer

Equations which are sums of exponential functions do not show analytical solutions in general.Your case seems among the particular ones since writing   y  =      e                  10        x             the euation writes      0.6          y      4        +  0.4  y  =  1which, unfortunately reduces to a quintic polynomial which is not the most pleasant to solve (few quintic equations have analytical solutions - if they have, they tend to show many radicals).Probably the easiest could be to look for the zero of  f  (  y  )  =  0.4      y    5    −      y    4    +  0.6If you graph the function, you could notice that the positive root is around 2.5. So, using Newton method for example      f    ′    (  y  )  =  2      y    4    −  4      y    3  gives the iterative scheme      y          n      +      1        =            8              y        n        5            −      15              y        n        4            −      3              10      (              y        n            −      2      )              y        n        3            Using       y    0    =  2.5, the interates will then be      y    1    =  2.46160000000000      y    2    =  2.45898401374040      y    3    =  2.45897234660929      y    4    =  2.45897234637796which is the solution for fifteen significant figures.Now, from y get x.

Seettiffrourfk6 · Accepted Answer

The solution of the equation can be expressed with radicals. Be ready for the monster  y  =      3    8    +      1    8        25    −                  16                           5                      2                          /                        3                                                            65                    −          5                3              +    8                  5                  (                      65                    −          5          )                    3        +      1    2              25      8        +                  5                  2                      /                    3                                                  65                    −          5                3              −          1      2                      5                  (                      65                    −          5          )                    3        +          195              8                  25          −                                    16                                             5                                  2                                      /                                    3                                                                                                      65                                −                5                            3                                +          8                                    5                              (                                  65                                −                5                )                                      3

solve the below equation with respect to x 0.6exp(−40/x)+0.4exp(10/x)=1

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New Questions in Calculus 1