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Find constants of function with given conditions

  1. Mar 6, 2014 #1
    1. The problem statement, all variables and given/known data
    Let ##a>0## and [itex]y(x)=\left\{\begin{matrix}
    -x ;& x<-a\\
    Cx^2+D;& -a<x<a\\
    x;& x>a
    \end{matrix}\right.[/itex]

    a) Find ##C## and ##D## so that ##y\in C^1(\mathbb{R})##
    b) For A>a calculate ##\int_{-A}^{A}(1-({y}')^2)dx##
    c) Is it possible to find ##C## and ##D## so that ##y\in C^2(\mathbb{R})##?


    2. Relevant equations



    3. The attempt at a solution

    Could somebody please check if there is anything ok?

    a)
    ##{y}'(x)=\left\{\begin{matrix}
    -1 ;& x<-a\\
    2Cx;& -a<x<a\\
    1;& x>a
    \end{matrix}\right.##

    Than ##{y}'(a)=-1=2Ca##, therefore ##C=\frac{1}{2a}##.

    We also know that ##y(a)=Ca^2+D=a## therefore ##D=\frac{a}{2}##.

    b)
    For ##A>a## and ##y(x)=\left\{\begin{matrix}
    -x ;& x<-a\\
    \frac{1}{2a}x^2+\frac{a}{2};& -a<x<a\\
    x;& x>a
    \end{matrix}\right.## the integral is

    ##\int_{-A}^{A}(1-({y}')^2)dx=\int_{-A}^{-a}(1-({y}')^2)dx+\int_{-a}^{a}(1-({y}')^2)dx+\int_{a}^{A}(1-({y}')^2)dx##

    First and last integral are both 0 ahile the second is ##\int_{-a}^{a}(1-({y}')^2)dx=\int_{-a}^{a}(1-(\frac{x}{a})^2)dx=\frac{16}{15}a##

    That's IF I didn't make a mistake...

    c)
    ##{y}''(x)=\left\{\begin{matrix}
    0;& x<-a\\
    2C;& -a<x<a\\
    0;& x>a
    \end{matrix}\right.##

    Everything suggests that ##C=0##, therefore the answer is NO.
     
    Last edited: Mar 6, 2014
  2. jcsd
  3. Mar 7, 2014 #2

    haruspex

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    I think you meant y'(a) = 1, not -1.
    I don't see how you get 16/15 in the final step in part b.
    Other than that, all looks good.
     
  4. Mar 8, 2014 #3
    Yes, I meant y'(a)=1.

    Thank you!
     
  5. Mar 8, 2014 #4

    haruspex

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    But what about the 16/15? That looks wrong to me.
     
  6. Mar 8, 2014 #5

    =) It is also wrong. The right result should be ##2a-\frac{2}{3}a=\frac{4}{3}a##.
     
  7. Mar 8, 2014 #6

    haruspex

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    Agreed.
     
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