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How do I solve the following equation?: $$ 0.2 = 0.819 \exp(-11.5x)+0.0975 \exp(-70.1x)+0.0325 \exp(-179x) $$ The result is supposed to be $x \approx 0.1225$

Zubzub
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Roberta L
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  • Most probably, there is no standard way for solving this type of question, by hand. If you want to know solution(s) only, you can use Wolfram Alpha. – Jaideep Khare Aug 17 '17 at 14:18

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The huge negative exponents will make the last two terms on the right very small, so you can get a preliminary value by ignoring them. We get $\ln \frac {0.2}{0.819}\approx -11.5x, x \approx 0.122588$ Now you can incorporate the other terms in a fixed point iteration $$0.2 = 0.819 \exp(-11.5x)+0.0975 \exp(-70.1x)+0.0325 \exp(-179x)\\ \frac {-1}{11.5}\log \left(\frac 1{0.819}\left(0.2-0.0975 \exp(-70.1x_i)0.0325 \exp(-179x_i)\right)\right)=x_{i+1}$$ The iteration will converge very rapidly

Ross Millikan
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