A businessman intends to buy a bungalow and he is going to get a home loan. He can afford to make payment of RM 𝑝 hundred thousand per year. The payments are distributed out and paid constantly throughout the year. The current interest rates are 𝑞%, compounded continuously. Assume that, the rate due to interest is proportional to the balance and the payments are removed from the balance at a constant rate. a) Let 𝑦(𝑡) be the loan balance after 𝑡 years. State the differential equation and solve to get 𝑦(𝑡). b) The businessman plans to get a 20-year loan. How much the home loan amount (in whole number) that he can obtain from the bank? c) Assuming the interest rate is fixed throughout the loan period, the businessman decided to make an advance payment of RM 𝑟 hundred thousand at Year 5. In which year will he settle the full payment of the loan?

p=5,q=1,r=6 Well, I don't understand Since y(t) is the loan balance after t years, then the rate of change of loan balance should be this clearer equation

  =q^2 y-pq-p
T2-a=q^2 y-pq-p-(qy-p)
           =q^2 y-pq-qy

T3=q^3 y-p(q^2+q+1)

T3-T2=q^3 y-q^2 y-pq
where a=loan balance first year, T2=loan balance second year, T3=loan balance third year, y= loan applied

Then I deduce that the y'(t)=q^t y-q^(t-1) y- pq,this clearer equation but seems like it is incorrect...maybe I should'nt deduce like that??


Does this help?

\begin{align} \frac{dy}{dt} &= -p + qy\\ \frac{dy}{qy-p} &= dt\\ \end{align} Integrating from $t=0$ to $t$ and $y=y_0$ to $y$, we get: \begin{align} \frac{dy}{dt} &= -p + qy\\ \frac{ln(qy-p)}{q}\bigg|_{y=y_0}^y &= t|_{t=0}^t\\ \frac{ln(qy-p)-ln(q y_0-p)}{q} &= t\\ ln(qy-p)-ln(qy_0-p) &= qt\\ ln(qy-p) &= qt+ln(qy_0-p)\\ qy-p &= (qy_0-p)e^{qt}\\ y &= \frac{(qy_0-p)e^{qt}+p}{q}\\ &=y_0 e^{qt} - \frac{p}{q}(e^{qt}-1) \end{align}

  • $\begingroup$ Hi, thanks for the answering. The answer you provided above is logical !! But, I still have a question about the amount of loan that the business will get from the bank is not given in this question, how can I know the total loan balance in 20 years?? I have built an equation about the total amount of loan in t(years), compounded with interest rate every year , x(t)=q^t x0, where x0 to be the initial loan applied. However, the x0 is not given. I wonder is there any hidden initial condition in this question? $\endgroup$
    – Pluto
    Apr 9 at 3:57
  • $\begingroup$ @Pluto, please see if this helps: $y(t) = y_0 e^{qt} - \frac{p}{q}(e^{qt}-1)$, then for t=20 $e^{-q20}(y(20) + \frac{p}{q}(e^{q20}-1))= y_0$ where $y_0,y(20)$ are the initial loan balance and balance after 20 years. But we repay loan fully in 20 years, so $y(20)=0$, so you can solve for $y_0$ from this equation $e^{-q20}(\frac{p}{q}(e^{q20}-1))= y_0$ for part 2. $\endgroup$ Apr 9 at 4:45
  • $\begingroup$ Thank you so much!! It helps!! I wonder is it possible I integrate the differential equation above in indefinite ways to get the same answer as u stated as above? $\endgroup$
    – Pluto
    Apr 9 at 9:54

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