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Differential Equation Model 
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#1
Apr312, 04:57 AM

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1. The problem statement, all variables and given/known data
Here's a model for the balance owed on a loan with the following conditions: * Interest accumulated on the loan at a rate of 5.24% per year * The amount owed at the beginning of the loan was $20,000. * No payments were made on the loan for the first two years * After two years, the loan was paid off at $3,000 per year. The model is given by the equations: [itex]\frac{dL}{dt} = 0.0524 \ L, \ \ \ 0<t<2[/itex] [itex]\frac{dL}{dt} = 0.0524 \ L \  \ 3 , \ \ t>2[/itex] (L is in thousands) Using a direction field work out how large repayments should be if the loan is to be paid back in exactly 12 years (i.e., with exactly 10 years of repayments after the first two years with no repayments). 3. The attempt at a solution Here is the direction field I made for this model And the solution for the initial value L(0)=20 is The root of this solution occurs at t=11.39, so 11 years is how long it will take to pay back the loan. But how can we work out how large repayments should be if the loan is to be paid back in 12 years? Any help is greatly appreciated. 


#2
Apr312, 12:42 PM

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Change the DE to dL/dt = 0.0524L  R for t > 2, and solve it with R as a symbolic parameter. Then find R that makes L(12) = 0.
RGV 


#3
Apr312, 04:11 PM

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Is this what you meant? 


#4
Apr312, 04:30 PM

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Differential Equation Model
RGV 


#5
Apr312, 04:56 PM

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dL/dt = 0.0524 L  0.6288 I'm not sure how this helps us. Did you mean I have to first solve the DE using separation of variables while ignoring R? I appreciate it if you could maybe explain that a bit more clearly. 


#6
Apr312, 06:14 PM

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It looks like you are using a phase plane plotter as well as the direction field. You can use it or the direction field to plot backwards as well as forwards in time! (And it looks like that is what your plotter is doing since your starting point is in the middle!)



#7
Apr312, 06:26 PM

P: 895

Yes I've used the Matlab addon called "dfield", my plot shows the initial condition y(0)=20. But how do can we use this plot to figure out how large repayments should be if the loan is to be paid back in exactly 12 years?



#8
Apr312, 07:19 PM

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I don't have access to Matlab, so I cannot offer technical advice. But if I were doing the question I would avoid the use of a phase plot and would, instead, just write down the formula for the DE solution; it would be a formula that has both R and t in it: L = f(t,R). Then I would set L=0 at t=12 and solve the equation to find R.
I guess if you want to use phase plot methods you could start by guessing a value of R, then make the phase plot for that R value and trace out the solution. If L=0 occurs before t=12 our guessed value if R is too large, so we should decrease it a bit and start over. If L=0 occurs after t=12, our guessed R is too small, so we should increase it a bit and start over. This procedure would be horrible and would take forever, and that is why I would not use it unless I had hours of spare time and nothing better to do. RGV 


#9
Apr312, 07:22 PM

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Your plot is actually predicting backwards from that t(0)  predicting your what your debt would have been over the previous 30+ years if your first eq. had been applicable over that time!
Now you know where your starting t is for the 'retrodiction'. t = 12. It should be obvious that at your present R = 3 the slopefield will not take you back from (12, 0) to (0, 20) i.e. L = 20 but to something higher. If not obvious make a few trials and you'll soon see. So as you cannot get back to 20 at the right time with that R try different R till it comes right! You can as RV said calculate it mathematically, which is not very hard depending on what math you know. But using the plotter is not even math, it's common sense. 


#10
Apr312, 07:24 PM

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#12
Apr312, 07:43 PM

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#13
Apr312, 07:49 PM

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You have all you need. (I have to go now.) 


#14
Apr312, 07:52 PM

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I really need more information from you in order to _guide_ you while avoiding doing your homework for you. What is the course? What is your background? How much calculus have you had? etc. RGV 


#15
Apr312, 08:01 PM

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#16
Apr312, 08:08 PM

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It may be that you are comfortable with solving the DE dL/dt = c*L but uncomfortable with the DE that has the extra 'R' on the right. Is that the case? If so, look instead at the DE for M = L  p, where p is some constant. You ought to be able to figure out what p value to choose in order to get rid of the constants on the right and have just dM/dt = c*M. Alternatively, you can use an integrating factor. RGV 


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