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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.

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  • $\begingroup$ or if i wanted to do it monthly i should as well change the rate to 0.01 $\endgroup$
    – Leavei
    May 8 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 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 at 8:19
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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.

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