Let y(t) represent your bank account balance, in dollars, after t years. Suppose you start with $10000 in the account. Each year the account earns 6% interest, and you deposit $5000 into the account. This can be modeled with the differential equation: dy 0.06y + 5000 dt y(0) = 10000 Solve this differential equation for y(t) y(t) =
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![### Differential Equation for Bank Account Balance with Interest and Deposits
#### Problem Statement:
Let \( y(t) \) represent your bank account balance, in dollars, after \( t \) years. Suppose you start with $10,000 in the account. Each year the account earns 6% interest, and you deposit $5,000 into the account annually.
This can be modeled with the differential equation:
\[ \frac{dy}{dt} = 0.06y + 5000 \]
\[ y(0) = 10000 \]
#### Solution:
Solve this differential equation for \( y(t) \):
\[ y(t) = \]
Explanation:
1. **Set-Up**: We start with an initial condition where \( y(0) = 10000 \). This means that at time \( t = 0 \), the account balance is $10,000.
2. **Interest**: The account earns 6% interest per year, which translates to a term \( 0.06y \) in the differential equation.
3. **Deposits**: You regularly deposit $5,000 into the account each year, hence the term \( +5000 \) is added to the differential equation.
#### Instructions:
1. Apply techniques for solving first-order linear differential equations.
2. Use the given initial condition to find the particular solution.
3. Provide the final solution in the provided form.
This differential equation models how the balance grows over time considering both interest and annual deposits.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62b092f3-63db-4d93-982f-67d0473d8e68%2F4e1f317b-f698-4d0e-b374-dd6662e07d66%2Fslxt9bc_processed.jpeg&w=3840&q=75)
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