Consider the two equations below: y 1 </msub> = ( 1 &#x22



Answered 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?

Answer & Explanation



Beginner2022-07-16Added 22 answers

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.


Beginner2022-07-17Added 6 answers

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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