Question

a continuous compounding question with diff. equation and initial conditions. don't know where to start. please help.

Answer 1

State your problem properly as it was stated in your hw!

Answer 2

You have $1000 with which to open an account which you plan also to add $1000 per year. All funds in the account will earn 10% annual interest compounded continuously. If the added deposits are also credited to your account continuously, the number of dollars x in your account at time t (yrs) will satisfy the initial value problem.
Diff. equeation dx/dt = 1000 + 0.10x
initial condition: x(0) = 1000

(a) solve the initial value problem for x as a function of t
(b) about how many yrs will it take for the amount in your acount to reach 100,000

Answer 3

\[
\frac{dx}{dt} = 1000 + 0.10x \\
\frac{dx}{1000+0.10x} = dt \\
\int \frac{dx}{1000+0.10x} = \int dt \\
\frac{\ln(1000+0.10x)}{0.10} = t \\
\ln(1000+0.10x) = 0.1t \\
(1000+0.10x) = e^{0.1t} \\
0.1x = e^{0.1t} - 1000 \\
x = 10e^{0.1t} - 10,000 + C \\
t = 0, x = 1000 \\
1000 = 10 - 10,000 + C \\
C = 10,990 \\
x = 10e^{0.1t} - 10,000 + 10,990 \\
x = 10e^{0.1t} + 990 \\
\]