Results 1 to 4 of 4

Math Help - Question about the constant that occures after differentiation/integration

  1. #1
    Newbie
    Joined
    Dec 2009
    Posts
    7

    Question about the constant that occures after differentiation/integration

    Hi!

    I have a question about something I find a bit peculiar. The textbook of my math-course is a bit inconistent in the use of the constant (here C) that comes from differentiating an expression.

    I have an expression that has been differentiated, and it looks like this:

    e^{\frac{1}{2}x^2+C}

    Sometimes this kind of expression is used to calculate further, but other times it is manipulated into this:

    Ce^{\frac{1}{2}x^2}

    My question is this:
    Can I always perfrom this manipulation, or is it just in certain circumstances? I know the constant C is a constant that can be changed if it is only affected by other constants (like e in this situation), but the whole thing is very confusing...

    Any clarifying help will be greatly appreciated!

    Thomas
    Follow Math Help Forum on Facebook and Google+

  2. #2
    -1
    e^(i*pi)'s Avatar
    Joined
    Feb 2009
    From
    West Midlands, England
    Posts
    3,053
    Thanks
    1
    Quote Originally Posted by luckytommy View Post
    Hi!

    I have a question about something I find a bit peculiar. The textbook of my math-course is a bit inconistent in the use of the constant (here C) that comes from differentiating an expression.

    I have an expression that has been differentiated, and it looks like this:

    e^{\frac{1}{2}x^2+C}

    Sometimes this kind of expression is used to calculate further, but other times it is manipulated into this:

    Ce^{\frac{1}{2}x^2}

    My question is this:
    Can I always perfrom this manipulation, or is it just in certain circumstances? I know the constant C is a constant that can be changed if it is only affected by other constants (like e in this situation), but the whole thing is very confusing...

    Any clarifying help will be greatly appreciated!

    Thomas
    As e is a constant and C is a constant it follows that e^C must be a constant (much like e^2 is)

    For exponential functions the laws of exponents can be used. In this case e^{f(x)+k} = Ce^{f(x)} where C = e^k

    I always denote the initial constant of integration with a different letter (usually k) and define what C is equal to at the end to avoid ambiguity
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Dec 2009
    Posts
    7
    Thank you!

    In my course we have been taught to "re-use" the C, but to explain how the C has changed on the right side of the expression where the "new" C first occures.

    I think we learn it this way because it seems easier (it is a beginners course).

    Thanks again, I am now more confident in my management of this constant.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    Threre is some controversials about the question. lets suppose to have the following DE...

    y^{'} -y=0 (1)

    ... which can be resolved in 'standard fashion' separating the variables as follows...

    \frac{dy}{y} = dx (2)

    Integrating both terms of (2) we have...

     \ln |y| = x + c (3)

    ... where c is an 'arbitrary constant' that we can write as c=\ln \chi, \chi>0. With exponentiation of both terms of (3) we finally obtain...

    |y|= \chi \cdot e^{x} (4)

    ... that is the 'general solution' of (1). Of course the (4) can be written as...

    y= \pm \chi \cdot e^{x} (5)

    ... and it covers all the possible IVP expressed by the (1)... with the only exception of the 'initial codition' y(0)=0 that has as solution y(x)=0 that doesn't correspond to any value of \chi because we have supposed \chi >0 ...



    Merry Christmas from Italy

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Constant of a series from Integration
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 25th 2011, 11:50 PM
  2. Question on differentiation/integration...
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 2nd 2010, 01:41 PM
  3. Replies: 6
    Last Post: July 21st 2010, 06:20 PM
  4. integration constant
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: May 25th 2010, 05:36 PM
  5. Replies: 1
    Last Post: June 30th 2008, 09:29 AM

Search Tags


/mathhelpforum @mathhelpforum