Where is the Mistake in Eq. 3?

[Pachito]

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

 

[JJacquelin]

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.