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$y$ is a function of $t$. Simple differential equations are written to make

$ \frac{dy}{dt} = ky(t) $

The function $y(t)$ that fits this is

$y(t) = y(0) e^{kt} $

Where $y(0)$ is some initial condition.

$ \frac{d}{dt} y(t) = k y(0) e^{kt} $

Achieving our constraint $ \frac{dy}{dt} = ky(t) $

This is all fine and dandy, but I'm wondering if anyone ever uses $2^{kt}, 3^{kt}$ or $n^{kt}$ in any problems. There is an accumulating $ln(2)$ factor on each derivative, but I'm wondering if that's ever useful.

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up vote 2 down vote accepted

It depends on $k$. If your $k$ is $\log(2)$, then you could write $y(t) = 2^t$.

Any exponential can be rewritten in terms of other exponential.

$a^{k_1 t} = b^{k_2 t}$, where $k_2 = k_1 \log_b(a)$ and equivalently $k_1 = k_2 \log_a(b)$.

But $f(t) = c e^t$ is the only "nice" function which satisfies $f'(t) = f(t)$

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One can always change bases and get back the exponential function and exploit the fact that $f(x) = f'(x)$.

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