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.

  • $\begingroup$ or if i wanted to do it monthly i should as well change the rate to 0.01 $\endgroup$
    – Leavei
    May 8, 2021 at 7:47
  • $\begingroup$ 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? $\endgroup$ May 8, 2021 at 8:15
  • $\begingroup$ The equation for b) should be $y'(t)=ry(t)+d\sum_{k=0}^\infty \delta(t-k·1month)$. $\endgroup$ May 8, 2021 at 8:19

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.