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# clase11handouts - Growth and Decay Phenomena Some standard...

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Growth and Decay Phenomena Some standard first-order differential equation models: The exponential growth: The population P is growing at a rate proportional to the population at any time t dP dt = kP , k > 0 . Thomas Malthus used this model to estimate the world population growth. The exponential decay: Let A be the amount of radioactive material in a sample at any time t . The amount A is decreasing at a rate proportional to the amount at any time t dA dt = kA , k < 0 . M.I. Bueno Differential Equations and Linear Algebra

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Both models have in common the linear differential equation y 0 - ky = 0 , which is homogeneous with constant coefficients. When k > 0, k is called the growth constant or rate of growth , and the equation is called the growth equation . When k < 0, k is called the decay constant or rate of decline , and the equation is called the decay equation . M.I. Bueno Differential Equations and Linear Algebra
Since this equation is linear and homogeneous, it is separable and, for y 6 = 0, the solution is given by y 0 y = k , dy y = kdt , ln | y | = kt + C , y = Ce kt , C R . If we consider the IVP, y 0 = ky , y ( t 0 ) = y 0 , then y 0 = Ce kt 0 , and the solution is given by y = y 0 e k ( t - t 0 ) . M.I. Bueno Differential Equations and Linear Algebra

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Growth and decay equations - Direction field. M.I. Bueno Differential Equations and Linear Algebra
Application The growth equation is a model useful in determining the future value of money. A deposit in a savings account earns interest, which is just a fraction of your deposit added to the total at regular intervals. The fraction is the interest rate and is based on a one-year period. An interest of 5 % means that 0 . 05 of the original amount is added after a year, so amount A grows to A + 0 . 05 = A ( 1 + 0 . 05 ) . M.I. Bueno Differential Equations and Linear Algebra

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The table gives the future value of an initial deposit A 0 , year by year, at an interest rate r = 0 . 05, compounded annually, Number Future value Future value of \$ 100 of years of account at 5 % annual interest 0 A 0 \$ 100 1 A 0 ( 1 + r ) \$ 100 ( 1 + 0 . 05 ) 2 A 0 ( 1 + r ) 2 \$ 100 ( 1 + 0 . 05 ) 2 . . . . . . . . . N A 0 ( 1 + r ) N \$ 100 ( 1 + 0 . 05 ) N M.I. Bueno Differential Equations and Linear Algebra
Suppose now that the bank pays interest n times per year. The fraction added in each period will be r / n . If the process is carried out for t years, the future value of the account will be A ( t ) = A 0 ( 1 + r n ) nt .

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clase11handouts - Growth and Decay Phenomena Some standard...

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