MATH103 April 2012
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Question 04 (e)
Consider the differential equation
where is a positive constant, t ≥ 0, k > 0, but y may be positive or negative. Suppose
y(0) = y0.
Suppose y0 = -k/2. Write the solution to the differential equation above in the form
What happens as ?
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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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Using similar reasoning as the previous problems,
When t=0, y(0)<0, so we need to take the positive branch outside the square root:
If we evaluate this at t=0, the positive branch gives , which is wrong, since y(0)=y0=-k/2. Hence this is the wrong branch, as before. The negative branch gives Our answer is therefore
As we have
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