Suppose a father wants to save $40,000 for a son\'s education over the next 15 y

Question

Suppose a father wants to save $40,000 for a son's education over the next 15 years. Suppose he opens a mutual fund that is earning about 5% per year. Let D be the amount he deposits into the fund in thousands of dollars per year. Let A(t) be the amount in thousands of dollars in the fund at time t. Assuming the father makes regular deposits so that the deposits over a 15 year period arc approximately continuous, the fund grows at a rate of A'(t) = r A + D. Solve the differential equation for A(t) assuming A(0) = 0. Determine D such that A (15) = 40 thousand. You should get D = $1, 790.51 per year. Determine the total amount the father deposits in the 15 years.

Explanation / Answer

From the given question,

A'(t)= rA +D

dA/dt= rA +D

dA / ( rA +D) = dt

integrating both sides,

1/r ln( rA +D) =t+c

replacing initial condition,

A(0)=0, when t=0, A=0, r=0.05

(1/0.05 ) ln( 0.05 x 0 +D) = 0 + c

20ln(D)=c ...............(1)

A(15)=40000, when t=15, A=40000

1/0.05 ln( 0.05 x 40000 +D) = 15 + c

20ln( 2000 +D)=15+c ...................(2)

20ln( 2000 +D)=15+20ln(D)

20ln( 2000+D) - 20ln(D)=15

20ln[(2000+D)/D]=15

ln[(2000+D)/D]=15/20

(2000+D)/D=e0.75

2000+D =e0.75D

2000/(e0.75-1)=D

D=$1790.51

Total amount father deposits in 15 years=$1790.51 x 15

=$26,857.6

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