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A concept from calculus that has always bothered me

  1. Jan 22, 2010 #1
    When one writes [tex]\int p(t) \, dt[/tex] is the constant of integration implied? I have always thought that it wasn't necessary to write [tex]\int p(t) \, dt +k[/tex]. However, in my diff. eq. book the constant is ussually written out, why is this so?

    In addition the book also writes [tex]\int_{t_o}^t p(t) \, dt +k[/tex] isn't the constant here unnecessary since we now have the limits of integration included?
     
  2. jcsd
  3. Jan 22, 2010 #2
  4. Jan 22, 2010 #3
    It's usually written because it's definite.

    Suppose

    [tex]
    y ' = f(x, y(x)) \ , y(x_0) = y_0
    [/tex]

    It does seem logical to say that (assuming f is integratable)

    [tex]
    \int y' dx = \int f(x,y(x)) dx \rightarrow y = \int f(x,y(x)) dx + k
    [/tex]

    but since we have initial conditions, we can do better.

    [tex]

    \int_{x_0}^{x} y'(t) dt = \int_{x_0}^{x} f(t,y(t)) dt \rightarrow y - y_0 = \int_{x_0}^{x} f(t,y(t)) dt
    [/tex]
    Thus,
    [tex]
    y = \int_{x_0}^{x} f(t,y(t)) dt + y_0
    [/tex]

    Better?
     
    Last edited: Jan 23, 2010
  5. Jan 23, 2010 #4
    Thanks l'Hopital
    Why is the later form preferred over the first form?
     
  6. Jan 23, 2010 #5
    Simply because it actually involves the initial conditions.
     
  7. Jan 27, 2010 #6
    There is one part in my book where it writes [tex] R'(y)=Q(x_0,y) [/tex]
    and then by integration [tex] R(y)=\int_{y_0}^y Q(x_0,y)\,dy[/tex].

    Shouldn't theauthor here have included a constant at the end of the integral as you have written in you example?
     
  8. Jan 28, 2010 #7

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Yes, because with that notation R(y0)= 0 which is not true for all anti-derivatives. I would also object to using "y" both as a limit of integration (and so outside the integral) and as the variable of integration. Much better would be either
    [tex]R(y)= \int_{y_0}^y Q(x_0,t)dt+ C[/tex]
    or
    [tex]R(y)= \int^y Q(x_0, t)dt[/tex]
    where we don't need the "C" because the lower limit of integration is left open.

    If you write just
    [tex]R(y)= \int Q(x_0,y)dy[/tex]
    the usual notation for the "anti-derivative", the constant C is implied. You do not write it there. Of course, if you wrote
    [tex]R(y)= \int y^2 dy= \frac{1}{3}y^2+ C[/tex]
    The constant on the right is necessary.
     
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