Where is the Mistake in Eq. 3?

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The discussion centers on the error in the integration of Equation 3 (Eq. 3) in relation to the numerical solution of Equation 1 (Eq. 1). The mistake identified is the incorrect assumption that exp(kt)dx is equivalent to d(x exp(kt)). The correct approach requires integrating exp(kt(x))dx, acknowledging that t is a function of x, rather than treating t as a constant. Consequently, Equation 4 (Eq. 4) is deemed false due to this oversight.

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Pachito
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Hello
everything follows up well, but when I compare the result (eq. 5) with the numerical solution of eq. 1, then are different. Where is the mistake? eq. 3?
thanks
 

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The mistake is in the integration of eq.[3] because
exp(kt)dx isn't equal to d(x exp(kt) )
So, Eq.[4] is false.
Do not forget that t is function of x.
t(x) is the reciprocal function of x(t)
So you would have to integrate exp(kt(x))dx, not simply exp(kt)dx with t constant.
 

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