Differential Equation (Deposit)

[mybingbinghk]

Homework Statement

A person initially place $500 in a saving account that pays interest at the rate of
4% per year compounded continuously. Suppose the person arranges for $10 per
week to be deposited automatically into the savings account.
(I) Write a differential equation for P (t), the amount on deposit after t years
(assume that “weekly deposits” is close enough to “continuous deposits” so
that we may model the balance with a differential equation.)

Homework Equations

Let X(t) be the amount on deposit after t years and dX/dt be the rate of change of the deposit.

The Attempt at a Solution

dX/dt = 0.04X +480 (480 is the amount of deposit yearly)
and
X(t) = C e^0.04t +480t , where C is a constant

Let X(0) = C =500, then
P(t)=500e^0.04t+480t

Acturally, I dun no the approach is correct or not? Pls Help!

 

[mighty2000]

Dear forum post author,

Your approach is mostly correct, but there are a few minor errors. First, the differential equation should be dP/dt, as P is the amount on deposit and not X. Secondly, the amount deposited weekly is $10, not $480 yearly. So the correct differential equation would be:

dP/dt = 0.04P + 10

Also, the initial deposit of $500 should not be included in the equation, as it is already taken into account when we set the initial condition P(0) = 500. So the final solution would be:

P(t) = 500e^0.04t + 10t

I hope this helps! Let me know if you have any further questions.