Consider the two equations below: y 1 = ( 1 − a 1 x ) e − α a 1 x y 2 = ( 1 − a 2 x ) e − α a 2 x Given y 1 , y 2 , a 1 and a 2 , is there an analytical way to determine α and x?

Question

Consider the two equations below:      y      1    =      (    1    −                  a              1            x        )        e          −                                    α                                      a                          1                                x                            y      2    =      (    1    −                  a              2            x        )        e          −                                    α                                      a                          2                                x                    Given       y      1  ,       y      2  ,       a      1   and       a      2  , is there an analytical way to determine   α and   x?

sniokd · Accepted Answer

A plot of the function reveals a solution close to x=0.01 and the solution, obtained using Newton, is x=0.0133409. From here, alpha=1.584997.I also used the solution based on the Lambert function. Here again, the plot of the fuction reveal a root close to alpha=1.5. Starting from here, Newton iterations lead again to alpha=1.584997. Frm here, x=0.0133409.Then, again, both approaches work and lead to the same results.Please post where are your problems. As told earlier, the problem is a resonable starting guess for the solution; this can be easily obtained looking at the plot of the function.

letumsnemesislh · Accepted Answer

Note that            y      1                  a        2                    y      2                  a        1              =                    (        1        −                              a                      1                    x                )                        a        2                            (        1        −                              a                      2                    x                )                        a        1            This eliminates   α.Not sure how you can solve this analytically. But once you get   x, you can solve for   α by taking logarithm of either equation.

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