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

Cristofer Watson 2022-10-21 Answered
solve the below equation with respect to x
0.6 exp ( 40 x ) + 0.4 exp ( 10 x ) = 1
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Answers (2)

Amadek6
Answered 2022-10-22 Author has 21 answers
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 = 1
which, 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 + 0.6
If 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 = 2.45897234637796
which is the solution for fifteen significant figures.
Now, from y get x.
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Seettiffrourfk6
Answered 2022-10-23 Author has 1 answers
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
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