Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

$y$ 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.

share|improve this question
add comment

2 Answers 2

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 $f'(t) = f(t)$

share|improve this answer
add comment

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.