I'm being asked the following question:
If $A(t)$ is the amount of the investment at time $t$ for the case of continuous compounding, write a differential equation satisfied by $A(t)$.
(The initial information given is that the initial value is \$2000 and the interest rate is 4%. I was previously asked to find the value of the investment at the end of 10 years if the interest was compounded at a varying number of times each year, which I did with no problem.)
When I completed the first part of the exercise, in the case of continuously compounded interest, I used the formula $A(t)=(2000)\cdot e^{(0.04)(t)}$
Referencing my textbook, I found a few notes that I think are relevant.
$y'(t) = C(ke^{kt}) = k(Ce^{kt}) = ky(t)$
Using this information, I tried writing the equation: $\frac{dA}{dt}=(0.04)(2000e^{0.04t})$ since the interest rate is k = 0.04, and C = 2000. This answer was not accepted however. What am I doing wrong?
Best Answer
You wrote it yourself, the differential equation is $$A'(t) = k A(t),$$ and all that remains is to pick out $k$. Of course, your solution is also a d.e. for $A$, it's just not interesting in that it doesn't express $A'$ in terms of $A$ alone. Indeed, the above d.e. shows that the relationship between $A'$ and $A$ is independent of time.