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A bank account earns $5 \%$ annual interest, compounded continuously. Money is deposited in a continuous cash flow at a rate of $\$ 1200$ per year into the account.$(a) Write a differential equation that describes the rate at which the balance $B=f(t)$ is changing.(b) Solve the differential equation given an initial balance $B_{0}=0$.(c) Find the balance after 5 years.

a. $1200+0.05 B(t)$b. $-24000+24000 e^{0.05 t}$c. $-24000+24000 e^{0.25} \approx \$ 6816.61$

Calculus 2 / BC

Chapter 9

Mathematical Modeling using Differential Equations

Section 5

Applications and Modeling

Differential Equations

Missouri State University

Harvey Mudd College

Baylor University

University of Nottingham

Lectures

01:11

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02:01

Money in a bank account ea…

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(a) If 3000 is invested al…

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Okay, So what we have here is a problem involving a money in a bank account That is compounded. Continue with me at 5% interest rate or 0.5 So money is also deposited at the cantina. Was cash flow amounting to $1200 per year. So we are required together V off the or not theory, let there be the particular solution. Even an initial contribution re of 060 and the money in the account off their five years. Okay, so the good be ISS, we can ram percent the money in the bank account as they be over the B is equal to 0.5 b plus 1200. We can barter isolate the 0.5 and you can represent the equation as 0.5 times be lost. But before 1000. So this is now a separable differential equation on, and we can rearrange our equation as the B over B plus 24,000 is equal to 0.5 bp, so integrating both sides will have, and in off B plus 24,000 is equal to zero point certified B Let's see, so we can get the power off the of both sides and we do this by doing here is to Ln off being plus 24,000 is equal the eeriest 0.5 b. Let's see so we can represent the left side US B plus 24,000 And remember that See, here is just a concept so it can think any value so we can write the right side. Us. CE raised to 0.5 p. So the further isolate the B on the left side. But then I started 24,000 r right side. So we'll have b is equal to see raised to 0.5 b minus 24,000. So this is our general solution on, but they want again. It's the particular solution for letter V. So for B of zero is equal to zero who just substitute this over equation and we'll have zero is equal to C E races, you know, 00.5 times zero my nose 24 out sent. So we know that this will just be equal to one and we'll have si is equal to going before said he started it. I think Arctic we should well have be off the Is Ecuador going before 1000 ce raised 0.5 b minus going before thou sent. So this is our particular. So they shot. Okay, so for another See, we want to find the amount off the money in the bank at these equal five? Nope. Sorry. So for leather, see, at the in support of five, we just substitute this to our particular solution And we have 24,000. Well, sorry. There should not be any See here. So our particular solution is done before 1000 erased 0.5 B minus with before 1000 G. So be off. Five will be equal to 24,000 e raise to 0.5 times five miners. 24,000. So computing for this this is approximately equal Cool. $6816.61. So this is our final answer

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