This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Solutions to Assignment 2  MAT1332A  Fall 2009 (1) (a) [6 points] Check that N ( t ) = t 1 + ct is a solution of the differential equation dN dt = N 2 t 2 . Treat c as an unspecified constant. (b) [5 points] Use N (1) = 1 to find c . Then give the solution N ( t ) corre sponding to this initial condition. Solution: (a) On one hand, dN dt = d dt t 1 + ct = 1 + ct ct (1 + ct ) 2 = 1 (1 + ct ) 2 , and on the other hand, N 2 t 2 = t 2 (1 + ct ) 2 1 t 2 = 1 (1 + ct ) 2 . Hence dN dt = N 2 t 2 . (b) N (1) = 1 1 + c = 1 =  1 c = 1 = c = 2 Therefore N ( t ) = t 1 2 t . (2) The rate at which a bacteria population multiplies is proportional to the instantaneous amount of bacteria present at any time. The mathematical model for this dynamics can be formulated as follows: db dt = kb, where b is a function in terms of the time t , b ( t ) is the number of bacteria at the time t , and k is a constant. The general solution of this autonomous differential equation is b ( t ) = b (0)...
View
Full
Document
 Fall '09
 ARIANEMASUDA

Click to edit the document details