Financial model using integral

In summary, the conversation discusses a financial model that involves continuously compounded interest and a continuous depletion rate. The problem is set up as a first-order differential equation, and the conversation explores different methods for solving it. The solution involves using partial fractions and integrating to find the amount in the savings account after one year, the maximum amount the account can grow, and how long it will take for the account to reach half of its maximum value.
  • #1
Maiko
2
0
financial model using integral

Find the amount in a savings aacount after one year if the initial balance in the account was $1,000, if the interest is paid continuously into the account at a nominal rate of 10% per annum, compounded continuously, and if the account is being continuously depleted at the rate of y^2/1000000 dollars per year, where y=y(t) is the balance in the account after t years. How large can the account grow? How long will it take the account grow to half this balance?

Just like other problems of this sort, I set up the following equation:
dy/dt=0.1y-y^2/1000000
integrating factor u(t)
dy/dt*u(t)=0.1y*u(t)-y^2/1000000*u(t)
d/dt(yu(t))=dy/dt*u(t)+du/dt*y
now, what do I do? I have never done a question involving y^2. Help, please!
 
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  • #2
After your first equation stop and solve the first problem. How large can the account grow? So er, when it is that large it isn't growing any more. :wink:

The second part,
dy/dt = ay - by2
no need for any integrating factors stuff.
It's something fairly simple of which you have probably done exercises with more complicated examples.
 
  • #3
Maiko said:
financial model using integral

Find the amount in a savings aacount after one year if the initial balance in the account was $1,000, if the interest is paid continuously into the account at a nominal rate of 10% per annum, compounded continuously, and if the account is being continuously depleted at the rate of y^2/1000000 dollars per year, where y=y(t) is the balance in the account after t years. How large can the account grow? How long will it take the account grow to half this balance?

Just like other problems of this sort, I set up the following equation:
dy/dt=0.1y-y^2/1000000
integrating factor u(t)
dy/dt*u(t)=0.1y*u(t)-y^2/1000000*u(t)
d/dt(yu(t))=dy/dt*u(t)+du/dt*y
now, what do I do? I have never done a question involving y^2. Help, please!

there is no 't' term on the right hand side. Don't blindly resort to one method. I am sure that your professors have taught you to use all of the tools available when solving a D.E.

What happens when you divide by [tex]0.1y-\frac{y^{2}}{1000000}[/tex]? seems like partial fractions to me.
 
  • #4
Your professors will also want you to learn the limitations of each method.

While every first order differential equation has, that particular method of finding an integrating factor only works for linear equations. That's why you never seen it with y2 before!

As epenguine and djeitnstine said, that is a separable equation. Separate the variables and integrate.
 

Related to Financial model using integral

1. What is a financial model using integral?

A financial model using integral is a mathematical representation of a company's financial performance over a period of time, typically using the integral calculus method. It takes into account various factors such as revenue, expenses, growth rates, and other financial metrics to provide a comprehensive analysis of a company's financial health.

2. How is integral calculus used in financial modeling?

Integral calculus is used in financial modeling to calculate the area under a curve, which represents the accumulated value of a company's financial performance. This allows for a more accurate and detailed analysis of a company's financial data, as it takes into account the changes and fluctuations in the data over time.

3. What are the benefits of using a financial model using integral?

A financial model using integral allows for a more precise and comprehensive analysis of a company's financial performance. It also helps in identifying potential risks and opportunities for growth, as well as making more informed financial decisions.

4. What are some common applications of financial modeling using integral?

Financial modeling using integral is commonly used in various industries such as banking, finance, and investment management. It can also be applied in project finance, mergers and acquisitions, and budgeting and forecasting.

5. How often should a financial model using integral be updated?

The frequency of updating a financial model using integral depends on the company's specific needs and the type of data being analyzed. However, it is recommended to update the model at least on a quarterly basis to ensure its accuracy and relevance.

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