Is the Solution to the Annuity Differential Equation Correct?

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SUMMARY

The discussion centers on solving the differential equation S' = k + rS, which models the accrued interest on an annuity with regular deposits. The proposed solution is S(t) = (So + k)e^rt - k, where So is the initial amount, k is the regular deposit, and r is the interest rate. Clarifications confirm that k remains a constant deposit amount, and the equation should not be modified to include k(t) as a variable dependent on time. The correct approach maintains k as a fixed value, leading to the conclusion that the differential equation should not be altered to S' = kt + rS.

PREREQUISITES
  • Understanding of differential equations, specifically first-order linear equations.
  • Familiarity with the concept of annuities and regular deposits.
  • Knowledge of exponential functions and their properties.
  • Basic grasp of initial conditions in differential equations.
NEXT STEPS
  • Study the derivation of solutions for first-order linear differential equations.
  • Learn about the financial mathematics of annuities and their formulas.
  • Explore the implications of varying deposit amounts in differential equations.
  • Investigate the use of Laplace transforms in solving differential equations related to finance.
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Mathematicians, finance professionals, students studying differential equations, and anyone interested in the mathematical modeling of annuities.

wakko101
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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.
 
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wakko101 said:
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.
 
thanks! but, another question, then...does that mean I have to start with the dif eq'n
S' = kt + rS ?

Cheers. =)
 

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