Question

Amka opens two bank accounts: a checkings account with balance b(t) and a savings account with balance s(t) , at time t (measured in years since opening the accounts). at the same time, amka deposits 1500 dollars into the checkings account and 700 dollars into the savings account. the bank is paying interest at continuous rate of 6% on the checkings account and 10% on the savings account. also, amka continuously transfers money from the checkings account to the savings account at the rate of half the balance of the checkings account. set up a linear system of differential equations of the form dbdtdsdt==m11b m12s,m21b m22s.

Answer 1

To represent the changes in Amka's checking and savings accounts with differential equations, we account for the interest earned and the continuous transfers between the accounts. This results in a system where db/dt = 0.06b - 0.5b and ds/dt = 0.10s + 0.5b. This models both the growth and the inter-account cash flow.

Step-by-step explanation:

Setting Up a System of Differential Equations for Bank Accounts

Considering Amka's financial situation, we must account for continuous interest growth and monetary transfers between accounts. The checking account has an initial balance of $1500 and grows at a continuous rate of 6%, while the savings account starts with $700 and grows at a rate of 10%. Moreover, Amka is transferring money continuously from the checking to the savings account at a rate of half the checking account balance.

Let's denote the balance of the checking account by b(t) and the balance of the savings account by s(t). We can then describe the changes in the accounts using differential equations. The differential equation for the checking account will be based on the interest it's earning and the amount being transferred to the savings account. The savings account equation will include the interest it's earning and the transfers received from the checking account.

The system of differential equations can be represented as:

• db/dt = (interest rate of the checking account) × b - (transfer rate to the savings account) × b
• ds/dt = (interest rate of the savings account) × s + (transfer rate from the checking account) × b

Substituting the given rates:

1. db/dt = 0.06b - 0.5b
2. ds/dt = 0.10s + 0.5b

These equations form the required linear system of differential equations.