Hye guys
Given $$a = b\ e^{mt}-c\ e^{kt},$$ where $a,b,c,m,k$ are constants.
I am trying to solve this implicit equation to find for $t$. Can anyone here suggest me any theorem/method which can help me to solve this?
Thanks
suggestion for implicit equation problem
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implicit-function-theorem
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0factorize exp(t) and take logarithm? – 2017-01-13
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0$t=\log(a/(b-c))$, or am I missing something ? – 2017-01-13
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0sorry guys, the equation is wrong. It should be a = b*exp(m*t)-c*exp(k*t) – 2017-01-13
1 Answers
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By setting $x=e^{mt}$, you can put the equation in the form
$$x^\alpha=px+q,$$
which is the intersection of a power law and a straight line.
In a few particular cases $\alpha=2,3,4,\frac12,\frac13,-1,-2,\cdots$ you can use the formulas for the polynomials up to quartic. But in general, there is no closed form, and you will need to resort to numerical methods.
You can discuss the number of real roots by changing the value of $q$ until the line is tangent to the curve, which occurs when
$$\alpha x^{\alpha-1}=p,$$ i.e.
$$x^*=\left(\frac p\alpha\right)^{1/(\alpha-1)}$$ and
$$q^*=\left(\frac p\alpha\right)^{\alpha/(\alpha-1)}-p\left(\frac p\alpha\right)^{1/(\alpha-1)}.$$
From one side to the other of $q^*$, the number of roots changes by two units.
