Solve the following differential equation and interpret the result?

Questions
1

Hello All

I ask Mathematica to solve the following DE:

DSolve[{T'[t] == 0.626161 * Log[10000.0/T[t]] * T[t],  T[0] == 200.0}, T, t]

It returns the message

Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.

and a solution (?)

{T -> Function[{t}, 10000. 2.71828^(-3.91202 2.71828^(-0.626161 t))]}

What does the message mean? Can I consider the expression returned as a solution or not?

Thanks a lot for your help

2

From Help page:

This message is generated when Solve and related functions use inverse functions to represent a solution.
Inverse functions normally give one branch of a general inverse, so equations that are solved in terms of inverse functions may omit solutions associated with other branches.

Exact values are better, because give no error messages.

DSolve[{T'[t] == 626161/1000000*Log[10000/T[t]]*T[t], T[0] == 200}, T[t], t]
 (*{{T[t] -> 
    2^(4 - E^(-626161 t/1000000)) 5^(4 - 2 E^(-626161 t/1000000))}}*)
3

Oh, … That is great. I didn’t think about this. Since I am looking for an exact solution, I have to enter exact values. That makes sense.

Thank you!