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?