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Hello Guys,

Basically my problem is that my teacher has taught everything in the chapter that dealth with differential equations, slop-fields, population models, and prey-predator models. However, she failed to teach us how to model different types of models. We only learned how to do the memorization model. For the test, we need to do one problem, and basically we need to model a differential equation for a loan and investment. We were given a letter and it reads:

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I am in the process of buying a house around here. It's my first time buying any real-estate, and I find the whole mortgage thing confusing. I agreed to pay $459,950 for the condo. I saved 1% and have 99% to go. I'll be taking a fixed rate mortgage. I called a few lenders, but they have no idea what there doing. They give me complicated tables and numbers to crunch, but it doesn't make sense.

I would like to understand what's going on. Your teacher helped explain a few things to me:

[A fixed rate 30-year mortgage is a 30 year loan with an annual interest rate that doesn't vary. You make payments every month, and the payments are determined so that it takes exactly 30 years to pay off the loan completely (amortization). Figuring our the monthly payments isn't simple, because you aren't just paying off the principal amount, you are also paying off any interest that accrues.

All loans are paid with interest, and the more frequently the interest is compounded, the mroe you have to pay. Mortgage loans are frequently compounded monthly, though there are some loans that are compounded daily. Even though you have to pay if your loan is frequently compounded, its not much. If you want to understand the amortization process, it's best to assume that the loan is compounded continuously (making time a continuous independent variable), and model the amount of the loan as a variable dependent on time by a differential equation that would relate the change in that amount to itself, taking into account the anuual interest and the monthly payments. You can use that differential equation to determine the monthly payments, and the assumption of continuity will overestimate it only a few pennies.]

I want to know the best thing to do with my money. The loan officers keep telling me to get a mortgage of 1 point which means given them all the money I saved up upfront. The yalso said the best fixed rate is 5.762%. However, I know this investor, thats says she can probably get me a 10.6% return on it; thats what her firm has been getting on the 30-year investment. There's also a 1.5% fee on the investment amount.

So I want to know whether getting a higher mortgage loan and investing the money is even worth the trouble in the long-run. Also, if i invest the money, I won't get the money back until 30 years from now. I also want to know how much higher my monthly payments will be for the mortgage loan.

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I understand from this that we need to create a differential equation, but HOW!! I keep thinking of the equation P=Ce^rt. It deals with issues like this, but it is not a differential equation. I also look at P=C(1+r/n)^nt, but that's not one either. The problem is I don't

know how to setup the differential equation for this, how to explain how i got it, and then to solve it and prove which decision is better. Me and the class are at a stand still. We are completely baffled. If one can, please help or guide me!!!

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# Differential Equation with loans and investments

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