# ODE, First ODE, Separable method, Linear First ODE, integrating factor

#### Teslathunder

##### 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=7, these are given values. These problems are related to ODE(first order ode, separable method/linear first ode, Integrating factor). I know how to solve ode, but i do not know how to applied ode to the problems. Pls help me, thank you.

#### HallsofIvy

##### Elite Member
Do you not know what a "first order ODE" is? It is any equation of the form $$\displaystyle \frac{dy}{dx}= F(x,y)$$.

In this problem, it is the balance left on the loan that is changing so that is y and dy/dt tells how it is changing. Now, why is the balance changing? It is being increased by the interest and decreased by the payments. "Assume that, the rate due to interest is proportional to the balance" so the interest is qy. The payments are p so the differential equation is
$$\displaystyle \frac{dy}{dt}= qy- p$$.

You say "p=1, q=2, r=7" but there is no "r" mentioned in the problem?

#### Teslathunder

##### New member
Hi, thanks for the help, r is in problem c). Can i ask, how do i find the C (constant value)?

#### Teslathunder

##### New member
Ignore the above question, i had solved a) and b) because of yr help. Can you help explain to me the 3rd question C)? thank you

#### jasmine122

##### New member
hi all, can u provide me the answer a,b and c? thanks a lot

#### RAW BEEF

##### New member
Hi Teslathunder, u mind showing me your equation for part a), i am too finding out about this question and got the equation of
p=2, q=4, r=5