Diff.eqn. help

  1. I need help figuring out the solution to this diff.eq.


    y(x) = x + (1/2)*∫(from u=-1 to 1)[ ( 1-| x – u | ) y(u) du] , x є [ -1, 1]

    I have to show that:
    y``(x) + y(x) = 0 , x є [ -1, 1]

    subject to:

    y(1) + y(-1) = 0

    y`(1) + y`(-1) = 2

    Thanks for any help you can give.
     
    Last edited: Apr 22, 2007
  2. jcsd
  3. What you mean is you need to prove that

    [tex]y(x)=x+\frac{1}{2}\int_{-1}^x (1-|x-u|)y(u)du ,\qquad x\in[-1,1][/tex]

    is a solution of [itex]y''(x)+y(x)=0[/itex] with the given boundary conditions or viceversa?

    If it is the frist one, you just have to remember the definition for absolute value is

    [tex]
    |x|=\left\{\begin{array}{rl} x & \hbox{if }x\ge 0, \\ -x & \hbox{if }x<0\end{array}\right.[/tex]

    So, the function [itex]y(x)[/itex] is actually

    [tex]y(x)=x+\frac{1}{2}\left\{\int_{-1}^x [1-(x-u)]y(u)du+\int_{x}^1 [1-(u-x)]y(u)du\right\} [/tex]

    Hence, all you have to do is use the fundamental theorem of calculus, and prove that it satisfies the boundary conditions.
     
    Last edited: Apr 22, 2007

  4. That is what I tried to do but for some reason I'm not coming up with the correct solution for the given boundry conditions...
     
    Last edited: Apr 22, 2007
  5. Are you deriving correctly?

    The reason I ask this is because you are not going to use a standard form of the fundamental theorem of calculus. For example, lets define the function

    [tex]F(x)=\int_a^x f(u,x)du.[/tex]

    By definition,

    [tex]F'(x)=\lim_{h \rightarrow 0} \frac{F(x+h)-F(x)}{h}=\lim_{h \rightarrow 0} \frac{1}{h}\left\{ \int_a^{x+h} f(u,x+h)du-\int_a^x f(u,x)du\right\}.[/tex]

    Rewrittig the right hand of the equation,

    [tex]F'(x)=\lim_{h \rightarrow 0}\frac{1}{h}\left\{\int_x^{x+h} f(u,x+h)du +\int_a^x \left[f(u,x+h)-f(u,x)\right]du\right\}.[/tex]

    If the function [itex]f(u,x)[/itex] is well behaved in the domain of definition (as yours is), then we can exchange the integral and the limit, and using the fundamental theorem of calculus,

    [tex]F'(x)=f(x,x)+\int_a^x \frac{\partial f}{\partial x}(u,x)du.[/tex]

    Now, you tell me what is the derivative of [itex]G(x)[/itex] when

    [tex]G(x)=\int_x^b g(u,x)du.[/tex]
     
    Last edited: Apr 23, 2007
  6. I have no idea... I thought I knew but now I don't..
     
  7. Remember to use your definitions.

    [tex]G'(x)=\lim_{h \rightarrow 0} \frac{G(x+h)-G(x)}{h}=\lim_{h \rightarrow 0} \frac{1}{h} \left\{\int_{x+h}^b g(u,x+h)du-\int_x^b g(u,x)du \right\}.[/tex]

    Rewritting the right side of the equality,

    [tex]G'(x)=\lim_{h \rightarrow 0} \frac{1}{h}\left\{\int_{x+h}^x g(u,x+h)du+\int_x^b g(u,x+h)du-\int_x^b g(u,x)du\right\}.[/tex]

    Can you take it from here?
     
    Last edited: Apr 23, 2007
  8. Or better yet, why dont you write [itex]G(x)[/itex] as

    [tex]G(x)=-\int_b^x g(u,x)du[/tex]

    and use my previous post?
     
  9. It's not completely done, but I got:

    y(x)= x + (1/2)*y(x) - (1/2)*y(x)*x^2

    does it look like I'm on the right track?

    using this I can get y``(x) + y(x) = 0
    and y(x) = (2*x) / (x^2 + 1)
    which will give me: y(1) + y(-1) = 0
    but I can't get the last condition to work out.

    I can't get y`(1) + y`(-1) = 0

    ????
     
  10. Do it carefully:

    [tex]y'(x)=1+\frac{1}{2}\left\{-\int_{-1}^x y(u)du+\int_x^1 y(u)du\right\}[/tex]

    and there you go. Calculating [itex]y''(x)[/itex] should be a piece of cake.
     
  11. how did you evaluate that and leave the integrals in? I evaluated the integrals and found an expression for y(x) and then took the deriv. and found y`(x). Can I not do it that way? I would like to see the method you used to work that out, if you dont mind. I have no idea how you did that.
     
  12. As I stated in my previous posts, you are not using the standar fundamental theorem of calculus, because the function you are trying to derive depends on [itex]x[/itex] on both the limits of integration, as on the integrand. Thats why there is an extra integral term (think of it as some sort of chain rule).

    The Fundamental Theorem of Calculus (roughly) states that

    [tex]\frac{d}{dx}\left(\int_a^x f(u)du\right)=f(x).[/tex]

    In your case, the integrand does not only depends on [itex]u[/itex], but also on [itex]x[/itex]. Thats why there is the extra term. In a previous post, I've proven that

    [tex]\frac{d}{dx} \left(\int_a^x f(u,x)du\right)=f(x,x)+\int_a^x \frac{\partial f}{\partial x}(u,x) du.[/tex]

    In your case, lets write [itex]y(x)[/itex] as

    [tex]y(x)=x+\frac{1}{2}\left\{\int_{-1}^x f(u,x)du +\int_x^1 g(u,x)du\right\},[/tex]

    where [itex]f(u,x)=[1-(x-u)]y(u)[/itex] and [itex]g(u,x)=[1-(u-x)]y(u)[/itex].

    Using my previous posts, the derivative of [itex]y(x)[/itex] is

    [tex]y'(x)=1+\frac{1}{2}\left\{f(x,x)+\int_{-1}^x\frac{\partial f}{\partial x}(u,x)du-g(x,x)+\int_x^1 \frac{\partial g}{\partial x}(u,x)du\right\}.[/tex]

    Substituting [itex]f,\,g,\,\frac{\partial f}{\partial x},\,\frac{\partial g}{\partial x}[/itex], you should come up with the correct answer.
     
    Last edited: Apr 23, 2007
  13. I did the exact same thing but I didnt have the f(x,x) and g(x,x) terms. what are f(x,x) and g(x,x)?
     
  14. HallsofIvy

    HallsofIvy 40,297
    Staff Emeritus
    Science Advisor

    They are exactly what they say: f(x,y) and g(x,y) with y replaced by x. For example if f(x,y)= (x+y)2, then f(x,x)= (x+ x)2= 4x2.

    That rather complicated formula AiRAVATA is giving you is a special case of Leibniz's formula:
    [tex]\frac{d}{dx}\left[\int_{\alpha(x)}^{\beta(x)} \phi(x,t)dt\right]= \phi(x,\beta(x))\frac{d\beta(x)}{dx}- \phi(x,\alpha(x))\frac{d\alpha(x)}{dx}+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial\phi(x,t)}{\partial x}dt[/tex]
     
  15. I think I'm getting close but I am stumped on int( u*y(u) du ) integrated from -1 to 1 ? The y(u) is throwing me off....
     
  16. You don't have to integrate anything, you have to diferentiate. In fact, you can't integrate as [itex]y(u)[/itex] is unknown.

    Lets recapitulate: if [itex]f(u,x)=[1-(x-u)]y(u)[/itex], then

    [tex]f(x,x)=y(x),[/tex]

    and

    [tex]\frac{\partial f}{\partial x}(u,x)=-y(u),[/tex]

    so

    [tex]\frac{d}{dx}\left(\int_{-1}^x [1-(x-u)]y(u)du\right)=y(x)-\int_{-1}^x y(u)du.[/tex]

    What is the derivative of the next term?
     
    Last edited: Apr 23, 2007
  17. For the second half to that part I get: - y(x) + int( y(u) du) from x->1

    so to find y``(x), what is the partial derivative for d [y(u)] / dx = ? wouldn't it be zero?
    I know I need to get: y``(x)= -y`(x) and d [y(u)] / dx = 0 wouldn't get me there. Do I still use Leibnitz rule to find y``(x) ?


    I also have another question... How do you use the symbolic notation?.. that would make this much easier.

    I appreciate the help.
     
    Last edited: Apr 23, 2007
  18. That is correct!, so

    [tex]y'(x)=1+\frac{1}{2}\left\{-\int_{-1}^x y(u)du + \int_x^1 y(u)du\right\}.[/tex]

    For the second derivative, using the fundamental theorem of calculus,

    [tex]\frac{d}{dx} \left(\int_{-1}^x y(u)du\right)=y(x),[/tex]

    (the other term is for you to calculate) and that's it!

    Click on the formulae to find out :)
     
    Last edited: Apr 23, 2007
  19. I see.... I think I have it now... I'll let you know how it comes out (now that you have helped me along)
     
  20. to verify the boundry conditions of:

    y(1) +y(-1) = 0

    don't I have to int( u*y(u) du ) integrated from -1 to 1 , found in the y(x) eqn.?
     
  21. I don't belive so, just evaluate and all terms will cancel out.
     
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