How can I solve this DiffEQ problem involving retirement savings and salary increases?

[mailman85]

I am having a lot of trouble solving this problem. I don't even know where to start. Any help would be greatly appreciated.

A 30 year old woman accepts an engineering position with a starting salary of $30000 per year. Her salary S(t) increases exponentially with S(t)=30e^(t/20) thousand dollars after t years. Meanwhile 12% of her salary is deposited continuously in a retirement account which accumulates interest at a continuous annual rate of 6%. a) Estimate change(A) in terms of change(t) to derive the differential equation satisfied by the amount A(t) in her retirement account after t years. b) Compute A(40), the amount available for her retirement at age 70.

 

[HallsofIvy]

A, the amount of money in the retirement account, changes in two ways: 1) she contributes money each year and 2) it earns interest.
The amount of money she contributes is 12% of her salary: 0.12S
and the interest is 6% of the amount in the account: 0.06A.

[DELTA]A= 0.12S+ 0.06A= 0.12(30exp(t/20)+ 0.06A

That's the amount each year. If [DELTA]t is a portion of the year, then each of these would be multiplied by [DELTA]t:

[DELTA]A= (0.12S+ 0.06A)[DELTA]t
= (0.12(30exp(t/20)+ 0.06A)[DELTA]t
[DELTA]A/[DELTA]t = 3.6 exp(t/20)+ 0.06A.

The differential equation is dA/dT= 3.6 exp(t/20)+ 0.06A or
dA/dt- 0.06A= 3.6 exp(t/20), a relatively straight-forward first order, non-homogeneous, linear equation with constant coefficients.

Assuming she started this retirement account when she started the job, then A(0)= 0 is the intial condition.

(I get that, after 40 years, her retirement account contains
1308.28330 thousand dollars or $1,308,283.30.) Might be enough to retire on!