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2051B06T1

# 2051B06T1 - du/dt 3 u = e 2 t 3 t where u = u(t 4 Let dy/dx...

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1 Prof. Tang MA2051 B06 Nov. 15, 2006 Test 1 Section____Name:______________________ Problem 1 2 3 4 5 Total Possible Points 25 20 20 10 25 100 Points awarded Write Clearly. Mark your solutions. Show Your Work 1. The total mass of a radioactive material can be modeled using a population model dm/dt= - km, m(0)=m 0 , (time unit: year; mass unit: gram; m 0 is the initial mass). a) (10%) Solve the model to obtain m(t). b) (10%) If the material loses half of its initial mass in 1400 years, find the k value. c) (5%) If m 0 =10,000 gram, how long does it take for m(t) to become 500 gram? 2. (20%) Solve: dy/dx + 2x y = x by using variation of parameters method.

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2 3. (20%) Solve:
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Unformatted text preview: du/dt + 3 u = e 2 t + 3 t where u = u(t). 4. Let dy/dx = 2x (y 2 +4). a) (5%) Is this equation linear? Why? b) (5%) Can it be solved by separation of variables? If no, Why? If yes, solve it. (hint: ∫ 1/(y 2 + a 2 )dy = 1/a arctan (y/a) + c). 3 5. a) (7%) A spring extends 9.8 cm when a 5 kg weight is applied to it. Find the spring constant. (SI unit system must be used. Gravity constant is g=9.8 m/s 2 ). b) (8%) Assuming no damping and no forcing, write down the complete model (initial value problem) for the spring-mass system given above. 5 kg is the mass. c) Let the initial conditions to be y(0)=0.1, y ′ (0)=0, solve the model....
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2051B06T1 - du/dt 3 u = e 2 t 3 t where u = u(t 4 Let dy/dx...

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