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theEric
 2 years ago
Hello! I'm working on a financerelated problem, and I'm not sure that I'm solving the ODE well!
Here's the prompt:
A certain college graduate borrows $8,000 to buy a car. The lender charges interest at an annual rate of \(10\%\). Assuming that interst is compounded \(\sf continuously\), and that the borrower makes payments \(\sf continuously\) at a constant annual rate \(k\), determine the payment rate \(k\) that is required to pay off the loan in \(3\) years. Also, determine how much interest is paid during the \(3\)year period.
The answers: \(k=$3086.64\ /\text{year}\); \($1259.92\)
theEric
 2 years ago
Hello! I'm working on a financerelated problem, and I'm not sure that I'm solving the ODE well! Here's the prompt: A certain college graduate borrows $8,000 to buy a car. The lender charges interest at an annual rate of \(10\%\). Assuming that interst is compounded \(\sf continuously\), and that the borrower makes payments \(\sf continuously\) at a constant annual rate \(k\), determine the payment rate \(k\) that is required to pay off the loan in \(3\) years. Also, determine how much interest is paid during the \(3\)year period. The answers: \(k=$3086.64\ /\text{year}\); \($1259.92\)

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theEric
 2 years ago
Best ResponseYou've already chosen the best response.0Here's what I did. Let \(D\) be the debt in dollars. Note that \(\dfrac{dD}{dt}=D'=.10Dk\implies D'.10D=k\). I'll use \(C\) variables as arbitrary constants. Then \(\Large {D=e^{\int .10dt}\int (k)e^{\int .10dt}dt\\~\\~\\ =k\ e^{\int .10dt}\int e^{\int .10dt}dt\\~\\~\\ =k\ e^{.10t+C_1}\int e^{.10t+C_2}dt\\~\\~\\ =k\ e^{.10t+C_1}(10 e^{.10t+C_2}+C_3)\\~\\~\\ =k\ e^{.10t}e^{C_1}(10 e^{.10t}e^{C_2}+C_3)\\~\\~\\ =k\ e^{.10t}C_4(10 e^{.10t}C_5+C_3)\\~\\~\\ =k\ e^{.10t}C_4(10) e^{.10t}C_5k\ e^{.10t}C_4C_3\\~\\~\\ =k\ e^0C_4(10) C_5k\ e^{.10t}C_4C_3\\~\\~\\ =10k\ C_6k\ e^{.10t}C_7\\~\\~\\ =k\ C_8k\ e^{.10t}C_7}\)

theEric
 2 years ago
Best ResponseYou've already chosen the best response.0Where might I be making a mistake? This doesn't look right... I see that I can simplify: \(\Large\qquad\qquad\qquad = k(C_8e^{0.10t}C_7)\)

anonymous
 2 years ago
Best ResponseYou've already chosen the best response.0I forget the reason behind it, but I'm pretty sure you just ignore the constant of integration when you find the integrating factor. I end up with \[D(t)=10k+Ce^{1/10~t}\]

theEric
 2 years ago
Best ResponseYou've already chosen the best response.0Okay! I solved on Wolfram Alpha and I think I got something like that! I have to check! But thank you! I was just using a formula that was derived with the integrating factor, I think. I don't quite understand it! :P

anonymous
 2 years ago
Best ResponseYou've already chosen the best response.0You're welcome! And by the way, the reason (according to wikipedia) is that we only need *a* solution, not the general solution, to the integral.

theEric
 2 years ago
Best ResponseYou've already chosen the best response.0Hmm! Thanks! I'll probably look into this tomorrow when I'm less tired! :) Thank you very much, @SithsAndGiggles !
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