Ok, I simply cannot get the answer that they have in the solutions manual, nor do they explain how they got the answer, so I really have no clue what I'm doing wrong.

The instructions for the problem are: Investment. A brokerage firm opens a new real estate investment plan for which the earnings are equivalent to continuous compounding at the rate of

. The firm estimates that deposits from investors will create a net cash flow of

dollars, where

is the time in years. The rate of change in the total investment

is modeled by

(a) Solve the differential equation and find the total investment

as a function of

. Assume that

when

.

(b) Find the total investment

*A* after 10 years given that

*P = $500,000* and

*r = 9%*.

Ok, so I rearranged the DE to be

and then determined that

and

. From there, I found

.

I then mutiplied each side by the IF to get

.

From there I doublechecked that the IF was correct, it was, and then I did

The integral on the left cancels with the derivative so

I then pulled P out as a constant to get

I then did parts, using

,

,

to get

From there I used

and

and since I only have a

I did

Solving the final integral I came up with

Dividing both sides by

to get

by itself, I got

Which is different from what the solutions manual got, which was

What am I doing wrong?