This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Growth and Decay Phenomena Some standard firstorder 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 > . 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 < . M.I. Bueno Differential Equations and Linear Algebra Both models have in common the linear differential equation y ky = , 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 y = k , dy y = kdt , ln  y  = kt + C , y = Ce kt , C R . If we consider the IVP, y = ky , y ( t ) = y , then y = Ce kt , and the solution is given by y = y e k ( t t ) . M.I. Bueno Differential Equations and Linear Algebra 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 oneyear period. An interest of 5 % means that 0 . 05 of the original amount is added after a year, so amount A grows to A + . 05 = A ( 1 + . 05 ) . M.I. Bueno Differential Equations and Linear Algebra The table gives the future value of an initial deposit A , year by year, at an interest rate r = . 05, compounded annually, Number Future value Future value of $ 100 of years of account at 5 % annual interest A $ 100 1 A ( 1 + r ) $ 100 ( 1 + . 05 ) 2 A ( 1 + r ) 2 $ 100 ( 1 + . 05 ) 2 . . . . . . . . . N A ( 1 + r ) N $ 100 ( 1 + . 05 ) N M.I. Bueno Differential Equations and Linear Algebra Suppose now that the bank pays interest n times per year....
View
Full
Document
This note was uploaded on 03/14/2010 for the course MATH 3C taught by Professor Jacobs during the Fall '08 term at UCSB.
 Fall '08
 JACOBS

Click to edit the document details