How Do You Solve an Annuity Differential Equation?

[_N3WTON_]

Homework Statement

Solve the annuity problem:
$\frac{dS}{dt} = rS + d$
$S(0) = S_0$

Homework Equations

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

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.

 

[ZetaOfThree]

At this point I am not sure what to do because I believe that the negative symbol should not be there...

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

 

[_N3WTON_]

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

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

 

[Mark44]

_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.

 

[Ray Vickson]

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

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 $V_0 = S_0 + d/r$.