Explanation Given: The initial amount deposited is $5,000. The interest is compounded continuously at a rate of 7%. And then money is continuously deposited at a constant rate of $2,100 a year. Formula used: If money is continuously deposited into the account at the rate of $ m per year, then A satisfies the equation d A d t − r A = m ; where A ( 0 ) = P where d A d t is the rate of change of the amount A , rA is the rate of growth from continuous compounding and m is the rate of deposit. Integrating factor: The integrating factor of a function f ( x ) is given by I ( x ) = e ∫ f ( x ) d x . Calculation: Given that the initial amount deposited is $5,000 into an account at a rate of 7% which is compounded continuously. And then money is continuously deposited at a constant rate of $2,100 a year. Then, substitute r = 0.07 , m = 2 , 100 and P = 5 , 000 in the above equation. Therefore, the amount A at any time t must satisfy d A d t − 0.07 A = 2 , 100 ; A ( 0 ) = 5 , 000 . Thus, the integrating factor is, I ( t ) = e ∫ − 0.07 d t = e − 0.07 t Multiply both sides of the standard form by the integrating factor. e − 0.07 t d A d t − 0.07 e − 0.07 t A = 2 , 100 e − 0.07 t ( A e − 0.07 t ) ′ = 2 , 100 e − 0.07 t [ ∵ ( x y ) ′ = x y ′ + x ′ y ] Integrating on both sides, ∫ ( A e − 0

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Chapter 9.3, Problem 63E

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To find: The amount in the account at time t, and the amount in the account after 5 years.

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Chapter 9.3, Problem 62E

Chapter 9.3, Problem 64E

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The amount in the account at time t , and the amount in the account after 5 years.

Solution Summary: The author calculates the integrating factor of a function f(x)=edisplaystyle int

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To find: The amount in the account at time t, and the amount in the account after 5 years.

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