Interest Modeling Problem Mistake

Question

Adriana opens a savings account with an initial deposit of $\$1000$. The annual rate is $6\%$, compounded continuously. Adriana pledges that each year, her annual deposit (deposited continuously) will exceed that of the previous year by $\$500$. How much will be in the account at the end of the tenth year?

First I calculated the annual deposit $D$, via $D' = D + 500, D(0) = 1000$, to find that $D = 1500e^t – 500$.

Then I calculated $P(10)$, where $P$ is the balance at time $t$, via $P' = .06P + 1500e^t – 500, P(0) = 1000$. However, this yields ~$\$35\ million$, which seems very high. I checked the ODE solutions on Wolfram, so my mistake must be in the modeling. What am I doing wrong?

Answer 1

Edit:

$D = 1000+500t$

$P' = 0.06P+1000+500t$

Solving this would give

https://www.wolframalpha.com/input/?i=p%27(t)+%3D+0.06p%2B1000%2B500t

$p(t) = c_1e^{0.06t} - 8333.33t -155556$

$p(0) = c_1 - 155556 = 1000$

$c_1 = 156556$

Now $p(t) = 156556e^{0.06t} - 8333.33t - 155556$

Now $p(10) = 46374$