# Annunity Equation Derivation

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1. Sep 21, 2014

### _N3WTON_

1. The problem statement, all variables and given/known data
Solve the annuity problem:
$\frac{dS}{dt} = rS + d$
$S(0) = S_0$

2. Relevant equations
Integrating factor method equation
Future value of an annuity equation (this should be the final answer):
$S(t) = S_0e^{rt} + \frac{d}{r}(e^{rt} - 1)$

3. The attempt at a solution
Ok, I am getting quite close to doing this derivation correctly. However, I keep ending up with a negative that should not be there.
First, I set:
$p(x) = -r$
Then:
$u(x) = e^{-rt}$
This means that I need to take the integral of:
$\frac{d}{dt} (e^{-rt}S(t)) = de^{-rt}$
After taking the integral of both sides I end up with:
$(e^{-rt} * S(t)) = - \frac{d}{r} e^{-rt} + C$
Therefore:
$S(t) = - \frac{d}{r} + Ce^{rt}$
At this point I am not sure what to do because I believe that the negative symbol should not be there. If somebody could point out where my mistake is I would greatly appreciate it.

Last edited: Sep 21, 2014
2. Sep 21, 2014

### ZetaOfThree

Why? Your answer seems to agree with $S(t) = S_0e^{rt} + \frac{d}{r}(e^{rt} - 1)$ if $C=S_0+\frac{d}{r}$.

3. Sep 21, 2014

### _N3WTON_

ok I see it now, I guess im just an idiot XD ...sorry for the waste of time, I forgot to do the initial value portion of the problem

4. Sep 21, 2014

### Staff: Mentor

_N3WTON_,
It's a good habit to get into to check a solution you get. If the solution you get 1) satisfies the initial condition, and 2) satisfies the differential equation, you're golden. You don't need us to verify that your solution is correct.

5. Sep 22, 2014

### Ray Vickson

An easier way is to note that if $V = S + (d/r)$ then $dV/dt = dS/dt = r V$, so $V(t) = V_0 e^{rt}$, where $V_0 = S_0 + d/r$.