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# sol_assign2 - Solutions to Assignment 2 MAT1332A Fall...

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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) e kt , as shown in class. (a) [5 points] Given that the initial population size b (0) doubles in two hours, find k . ( Hint: use the logarithmic function )

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sol_assign2 - Solutions to Assignment 2 MAT1332A Fall...

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