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Constant Function

  1. Jun 1, 2008 #1
    1. The problem statement, all variables and given/known data
    Data Given:
    f'(x)=f(x) ..........(i)
    f(0)=0 ..........(ii)

    What kind of function is f(x)?

    3. The attempt at a solution
    From (i)
    1/y dy= dx
    ln y = x+C
    From (ii)
    ln 0=0+c
    Therefore; c is not defined!

    My book gives the answer that f(x) is a constant function. But how is it true? Is my way of solving wrong?
    Please Help
  2. jcsd
  3. Jun 1, 2008 #2


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    Homework Helper

    From the part in bold

    then use your initial values.
  4. Jun 1, 2008 #3


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    You divided by y when y = 0. I don't think you're supposed to solve it, but rather argue why it can't be anything other than a constant function based on the equation and the initial values.
    Last edited: Jun 1, 2008
  5. Jun 1, 2008 #4
    y=any constant clearly doesn't satisfy (i)

    though y=0 does satisfy..i think the answer is y=0 not y=any constant

    when you do dy/y=dx, you are assuming that y is not equal to zero, because dy/y will not be defined for y=0

    So, you have to consider the case y=0 separately
  6. Jun 1, 2008 #5
    The solution of your ODE should give y = Ae^x (for some constant A)
    Applying your initial condition gives 0 = Ae^0 ==> A = 0

    If I'm right the function y = 0 is just as much a constant function as say the function y = 5
    Last edited: Jun 1, 2008
  7. Jun 2, 2008 #6


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    No, integrating does NOT give "ln y= x+ C". Integrating gives ln |y|= x+ C. That's your crucial error.

    To see why that is important, take the exponential of both sides.
    eln |y|= ex+ C or |y|= C' ex where C'= eC. While eC must be positive, we can drop the absolute value by allowing C' to be negative as well, and, by continuity, C'= 0.

    y(x)= C'ex obviously satisifies the given differential equation for any real number C'.

    Now, put y=0, x= 0 into y= C'ex. What is C'? What is y(x)?
  8. Jun 2, 2008 #7
    Data Given:
    f'(x)=f(x) ..........(i)
    f(0)=0 ..........(ii)

    What kind of function is f(x)?

    f(x) is a constant function..... f(x) = 0
    where f'(x) = f(x) holds true. (The only other possibility was f(x) = e^x but then f(0) is not 0.)
  9. Jun 2, 2008 #8
    From your attempt at the solution you just get that c = 0.... but that doesn't help really.
  10. Jun 2, 2008 #9


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    There is a simple solution f(x) = 0. What do you know about the uniqueness of the solution?
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