# Modelling interest with differential equations (IVP)

Problem : you set a bank account, with initial value k, the bank will pay you continuous interest of 12% per year.

a) write an initial value problem for your account balance y(t) after t years

Sol: $$dy/dt = 0.12y$$ ik how to solve the IVP

b) Suppose that you will be continuously deposit $1000 per month into your account, write the IVP for your account balance y(t) after t years sol: i'm not sure here should i just multiply the 1000 by 12 to get it in years $$dy/dt = 0.12y + 12000$$ but doesn't this mean that there's no difference between puting$1000 each month

or he puts $12000 as a whole in a year ? last: it says that one of the solutions of the DE in part b is constant, Find it and what is the real life meaning of that constant. • or if i wanted to do it monthly i should as well change the rate to 0.01 May 8, 2021 at 7:47 • Is the 12% (absolutely unrealistic, either a direct fraud or very risky) the nominal or the effective rate? Why go to differential equations for a problem that can be solved via simple recursion? May 8, 2021 at 8:15 • The equation for b) should be$y'(t)=ry(t)+d\sum_{k=0}^\infty \delta(t-k·1month)$. May 8, 2021 at 8:19 ## 1 Answer This question leaves a bit up to interpretation so I shall address it. When I read part (b), I picture that the account holder is making continuous deposits (in the same manner as continuously compounded interest) over the course of each month that amounts to a thousand dollars each month. In such a case, your ODE $$\frac{\text{d}y}{\text{d}t} = 0.12y + 12000$$ will correctly model the situation. If, however, the account holder makes a single deposit each month for a thousand dollars then we model it as Lutz Lehmann suggests as $$\frac{\text{d}y}{\text{d}t}=0.12y+1000\sum_{k=0,1}^{\lfloor b\rfloor} \delta\bigg(t-\frac{k}{12}\bigg) \text{ .}$$ Here, $$\lfloor b\rfloor$$ is the floor of the time in months that we are considering the ODE and $$\delta$$ is of course the delta function. It should be noted that the sum should begin at $$k=0$$ if the account holder makes a deposit at $$t=0$$ and should start at $$k=1$$ if the account holder waits a month before making the deposit. So just to clarify your question doesn't this mean that there's no difference between puting \$1000 each month or he puts \\$12000 as a whole in a year ?

the ODE you wrote assumes that he is constantly making deposits that amount to a thousand per month (or equivalently twelve-thousand per year). If they are distinct deposits, it will of course make a difference when the deposits happen as they will be modeled by a delta function for every deposit.

As for the follow-up question it simply means that (in ODE speak) the particular solution to the non-homogenous problem is a constant function. In other words, you will get a solution that looks like $$y(t) = f(t) + B$$ for some constant $$B$$. I'll leave its physical interpretation up to you once you solve your ODE.