February 12

section covered in text
2.2  Separable Variables (conclusion)

solved problems - using key ideas
solving initial value problems, #21
possible loss of solutions when dividing by p(y)
formal justification for method, page 43
solutions not exprseeible in terms of elementary functions, #27b
mixture problem, #33
 
 

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QUESTION OF THE DAY
Let dy/dx = 1/x.
Where is the solution increasing? decreasing?
Where is the solution concave up? down?
Do any solutions intersect?
  SOLUTION

 

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SOLUTION. 
From calculus we know that a function is increasing when its first derivative is positive.  Therefore, this function is increasing on the interval (0,¥) , and decreasing on the interval (-¥,0).
By the second derivative test we know that the function is concave down when the second derivative is negative.  Since the second derivative is -1/x2 the solution is concave down on both intervals.
By Theorem 1 the solutions are unique on the intervals (0,¥) and (-¥,0).

 
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