Question

continous compounding you have $1000 wiht which to open an account to which you plan also to add $ 1000 per year. all funds in the account will earn 10% annual interest compounded continously. if the added deposits are also credited to your account continously, the number of dollars x in your account at time t(years) will satisfy th einitial calue problem.

differentital equation

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 years will it take for the amount in your account to reach $ 100,000

[I know how to plug in for part , but can anyone help me on the part A to find t]

Answer 1

Use the method of separation of variables to solve the differential equation, and the initial condition x(0) = 1000 to get the cosnstant term. The resulting x(t) will have an exponential form.

Separation of variables will involve the integration of

dx/(1000+0.10x) = dt

The resulting equation will be an approximation that is valid only when t is an integral number of years. During the interim, the balance will be a stepwise function until the next deposit and interest payment is made.