CK-12-Calculus

7.7. Ordinary Differential Equations http://www.ck12.org

Solution. The solution is given byP= 1 −AeP^00. 05 twhereA=^50001000 −^1000 =4.

P( 20 ) = 1 + 45000 e− 0. 05 ( 20 )= 1 +^50004 e− 1 = 2023

P( 30 ) = 1 + 45000 e− 0. 05 ( 30 )= 1 +^50004 e− 1. 5 = 3785.

Solve for time, 4000= 1 + 45000 e− 0. 05 (t)givese(−^0.^05 t)=

(^50004000) − 1
4 =^0 .0625. Sot=56. The population first exceed 4000 in
the 56thyear.
Exercise

1. (Exponential Growth) The population of a suburban city increased from 10000 in 2005 to 30000 in 2007.
   Assuming an exponential growth model on the population, by which year will the population first exceed
   100000?


2. (Logistic Growth) The population of a city is given by the equationdPdt = 0. 06 P( 1 − 100000 P^0 ),P 0 =25000. Find
   the population sizesP( 10 ),P( 25 ). At what time will the population first exceed 90000?

Multimedia Links

For a video presentation of Differential Equations including growth and decay(27.0), see Differential Equations,
Growth and Decay (7:23).

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/613

Numerical Methods (Euler’s, Improved Euler, Runge-Kutta)

The Euler’s method is a numerical approximation to a solution curve starting from the point(a,b)through the
algorithm:
yn+ 1 =yn+hF(xn,yn)wherex 0 =a,y 0 =bandhis the step size.
The shorter step size, the better is the approximation to the solution curve.
Improved Euler (Heun) method adapts on Euler’s method by using both end point values:yn+ 1 =yn+h 2 [F(xn,yn)+
F(xn+ 1 ,yn+ 1 )].
Sinceyn+ 1 also appears on the right side, we replace it by Euler’s formula,

yn+ 1 =yn+h 2 [F(xn,yn)+F(xn+ 1 ,yn+hF(xn,yn))].