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Financial model using integral 
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#1
Feb809, 03:47 PM

P: 2

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=0.1yy^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!! 


#2
Feb809, 05:28 PM

HW Helper
P: 1,967

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.
The second part, dy/dt = ay  by^{2} no need for any integrating factors stuff. It's something fairly simple of which you have probably done exercises with more complicated examples. 


#3
Feb1409, 02:13 AM

PF Gold
P: 619

What happens when you divide by [tex]0.1y\frac{y^{2}}{1000000}[/tex]? seems like partial fractions to me. 


#4
Feb1409, 06:07 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,357

Financial model using integral
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 y^{2} before! As epenguine and djeitnstine said, that is a separable equation. Separate the variables and integrate. 


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