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Samuelb88
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Homework Statement
A college graduate borrows $100,000 at an interest rate of 9% to buy a house. The college graduate plans to make payments at a monthly rate of 800(1+t/120), t is the number of months since the loan was made. Assuming this payment schedule can be maintained, when will the loan be fully paid?
The Attempt at a Solution
Let S(t) be the amount to be paid at some t
S(0) = 100,000
r = interest rate = .09
k = constant payment rate = -(800 + 20t/3)
So the differential equation modeling this problem is:
(i) [tex] \frac{dS}{dt}\right = rS +k[/tex]
The equation is linear and its integrating factor is u = Exp[-rt] so multiplying eq. (i) through by u, rewriting LHS by as the derivative of the product S(t)*Exp[-rt], and integrating across t:
(ii) [tex]S(t)*e^-^r^t=\int-e^-^r^t(800+20t/3)dt + C[/tex]
Integrating the RHS integral by parts letting U = 800+20t/3, dV = -Exp[-rt]:
(iii) S(t)*Exp[-rt] = (1/r)*(800+20t/3) - (20/3r) \int Exp[-rt]dt + C
Evaluating the last integral, and multiplying eq. (iii) through by Exp[rt]:
(iv) S(t) = (1/r)(800+20t/3) + (20/3r^2) + CExp[rt]
Exploiting the initial condition, C = 100,000 - (1/r)(800+20t/3) - (20/3r^2), so:
S(t) = (1/r)(800+20t/3) + (20/3r^2) + (100,000 - (1/r)(800+20t/3) - (20/3r^2))Exp[rt]
So the loan should be paid at some time t` when S(t`) = 0. Since the expression for S(t) has arguments of t as a scalar multiple and raised as an exponent, I solved this problem numerically. Using mathematica, it spits a value of t =~ -131.12 months. The answer provided by my book says t =~ 135.36 months.
I'm pretty stumped on this problem... Any help would be greatly appreciated.
PS Sorry latex doesn't seem to be working for me.
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