Science:Math Exam Resources/Courses/MATH103/April 2012/Question 04 (d)
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Question 04 (d) 

Consider the differential equation where is a positive constant, t ≥ 0, k > 0, but y may be positive or negative. Suppose y(0) = y_{0}. Suppose y_{0} = k/2. Write the solution to the differential equation above in the form What happens as ? 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Plug in into the solution of Problem 4b. Remember that and use this to select the appropriate signs in the expression. Then take the limit as . 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Using similar reasoning as the previous problem, When t=0, y(0)>0, so we need to take the positive branch outside the square root: again the last equality holding since k is positive. If we evaluate this at t=0, the positive branch gives , which is wrong, since y(0)=y_{0}=k/2. Hence this is the wrong branch. Indeed, the negative branch gives Our answer is therefore As we have that 