# The rate of change of Loan Balance

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

a=qy-p
T2=(qy-p)(q)-p
=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??

\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}
• @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. Apr 9 at 4:45