Is the Solution to the Annuity Differential Equation Correct?

[wakko101]

I'm trying to figure out the solution to a differential equation that describes the accrued interest on an annuity - rather than a lump sum at the beginning, we're dealing with multiple and regular deposits. The prof implied that we would have to solve for the dif eq'n
S' = k + rS
where k is the deposit, r the interest rate and S(t) the amount of money accrued at time t. I think I've solved for this particular equation (with intinial condition S(0) - So) which is
S(t) = (So + k)e^rt - k
but I have two questions. First, this doesn't seem right in terms of the fact that I'm subtracting k at the end there. And secondly, should k somehow be in terms of t as well (i.e. k(t) since the deposits are regular, i.e. at time t, I will have made k(t) deposits?)

Any insight would be appreicated...

Thanks,
W.

 

[HallsofIvy]

I'm trying to figure out the solution to a differential equation that describes the accrued interest on an annuity - rather than a lump sum at the beginning, we're dealing with multiple and regular deposits. The prof implied that we would have to solve for the dif eq'n
S' = k + rS
where k is the deposit, r the interest rate and S(t) the amount of money accrued at time t. I think I've solved for this particular equation (with intinial condition S(0) - So) which is
S(t) = (So + k)e^rt - k
but I have two questions. First, this doesn't seem right in terms of the fact that I'm subtracting k at the end there.

Look what happens when t= 0. e^0= 1 so S(0)= S0+ k- k. that's why you have to subtract off k- initially, you don't deposit k dollars at t= 0.

And secondly, should k somehow be in terms of t as well (i.e. k(t) since the deposits are regular, i.e. at time t, I will have made k(t) deposits?)

No! k is the amount of money you deposit each month and that is a constant. At time t (months) you will have kt (k times t) not k(t) (k of t).

Any insight would be appreicated...

Thanks,
W.

 

[wakko101]

thanks! but, another question, then...does that mean I have to start with the dif eq'n
S' = kt + rS ?

Cheers. =)