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Initial value problem - differential equations

  1. Dec 6, 2016 #1
    1. The problem statement, all variables and given/known data
    I am given (y^2 + y sin x cos y) dx + (xy + y cos x sin y) dy = 0, y(0) = π/2 .
    I need to solve this
    2. Relevant equations


    3. The attempt at a solution
    upload_2016-12-6_20-16-41.png

    At this point they still aren't exact, so I gave up. I can't figure out what the problem is. Is it possible that I have to continue trying to make them exact?

    I know that the answer is: xy-cosxcosy=0
     
  2. jcsd
  3. Dec 6, 2016 #2

    Mark44

    Staff: Mentor

    Divide both sides by y, to get ##(y + \sin(x)\cos(y))dx + (x + \cos(x)\sin(y)) dy = 0##.
    Now check for exactness. Note that since ##y(0) = \pi/2##, it's reasonable to assume that ##y \ne 0##.
     
  4. Dec 6, 2016 #3
    Can we always do this if we have a common factor like y in this case?
    Thank you, I spent a very long time trying to figure this out
     
  5. Dec 6, 2016 #4

    Mark44

    Staff: Mentor

    I wouldn't say you can always do it. By getting rid of a factor of y, we are possibly dividing by zero. Since the initial condition is that ##y(0) = \pi/2##, if necessary, we can restrict y to some interval that includes ##\pi/2## but doesn't include zero.
     
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