How Can You Model a Continuously Compounding Savings Account with Depletion?

[Maiko]

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!

 

[epenguin]

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 - by²
no need for any integrating factors stuff.
It's something fairly simple of which you have probably done exercises with more complicated examples.

 

[djeitnstine]

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 0.1y-\frac{y^{2}}{1000000}? seems like partial fractions to me.

 

[HallsofIvy]

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.