# Differential Equation (ODES)

#### jasmine122

##### New member
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=1, q=2, r=3, these are given values. These problems are related to ODE(first order ode, separable method/linear first ode, Integrating factor).THANK YOU FOR UR HELP

#### HallsofIvy

##### Elite Member
y(t) is the amount left to pay after t years. dy/dt is the rate of change of that amount. There are two things that change that amount. First there is a payment of 100,000p per year. That will decrease the amount owed so is negative. Second there is the interest. The interest rate is q% annually so the interest will be qy/100 each year. That increases the amount owed so is positive.

The differential equation is dy/dt= qy/100- 100,000p or dy/dt= (qy- 10000000p)/100.
We can separate variables as dy/(qy- 10000000p)= dt/100.

To integrate the left side, let u= qy- 10000000p.

With p= 1 and q= 2 the equation is dy/(2y-10000000)= dt/100.

If u= 2y- 10000000 the du/dt= 2 dy/dt so dy/dt= (1/2)(du/dt) so the equation is
(1/2) du/u= dt/100. That should be easy to integrate.

• jasmine122