Let's examine if the two functions provided are equivalent:

Function 1:

$x(t) = \frac{\frac{10}{9} ce^{10t}}{1 + \frac{1}{9} ce^{10t}}$

Function 2:

$x(t) = \frac{10}{1 + 9e^{-10t}}$

We will simplify both expressions and check for equivalence.

Step 1: Simplifying Function 1

$x(t) = \frac{\frac{10}{9} ce^{10t}}{1 + \frac{1}{9} ce^{10t}}$ Multiply the numerator and denominator by 9 to eliminate the fractions: $x(t) = \frac{10ce^{10t}}{9 + ce^{10t}}$

Step 2: Simplifying Function 2

$x(t) = \frac{10}{1 + 9e^{-10t}}$ Multiply numerator and denominator by $e^{10t}$: $x(t) = \frac{10e^{10t}}{e^{10t} + 9}$

Step 3: Comparing the two simplified forms

Function 1: $x(t) = \frac{10ce^{10t}}{9 + ce^{10t}}$

Function 2: $x(t) = \frac{10e^{10t}}{9 + e^{10t}}$

The expressions look similar, but they will only be equal if $c = 1$.

Thus, the two functions are equivalent only if $c = 1$.

Would you like further details or have any questions on this? Here are five related questions to consider:

1. How do constants affect the shape of an exponential function?
2. What happens to the behavior of $x(t)$ as $t \to \infty$?
3. Can you manipulate the exponentials in a more general case for different constants?
4. How does simplifying exponentials help in identifying equivalencies?
5. What are some real-world applications of such functions?

Tip: Always check if there are hidden constants or factors before concluding that two functions are identical.