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Help me with this

  1. Feb 12, 2007 #1
    Hello!!!
    My English is poor because I'm from Poland.

    I have 3 exercises from subject called: Mathematical Method In Physics.
    Would you be so kind and help me to solve this:

    1.
    (a) Prove that: [tex]{1 \over r}(rV)''=V''+(2/r)V' [/tex](that was easy, but next point (b) is not)
    (b) Use equation from (a) to solve:
    [tex] u''+{2 \over r}u'=1[/tex], where 1<r<2, u(1)=1, u(2)=2

    2.
    With witch k border problem:
    -u''-u=ku on (0, 1), u'(0)=1=u'(1)
    have solution?

    3.
    Find restricted solution of border problem:

    [tex]\Delta u-u=|x|^2[/tex] where [tex]x \in [/tex]B1(0) tight in R^3
    u(x)=1 when |x|=1, |x| is euklides length of vector.
    Use spherical variables and find solution dependent on "r"
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 12, 2007 #2

    berkeman

    User Avatar

    Staff: Mentor

    Welcome to the PF, ywel84. One of the rules here is that you must show your own work so far, before we can help you out. We do not provide answers here, but we can provide helpful hints and so forth, after you fill in your answers for questions #2 and #3 in the Homework Help Template above.
     
  4. Feb 12, 2007 #3

    cristo

    User Avatar
    Staff Emeritus
    Science Advisor

    What have you tried? Have you used part (a)?

    Hint: This can be rewritten as u''+(k+1)u=0
    Again, what have you tried?

    Please note that for homework questions, we must see your work before we can help. Also, in future, please post in the homework forums!

    edit: damn, beaten to it!
     
  5. Feb 12, 2007 #4
    For first exercise:
    1. The problem statement, all variables and given/known data
    (a) Prove that: [tex]{1 \over r}(rV)''=V''+(2/r)V' [/tex]
    (b) Use equation from (a) to solve:
    [tex] u''+{2 \over r}u'=1[/tex], where 1<r<2, u(1)=1, u(2)=2

    2. Relevant equations //I don't understand this
    r'=1
    r''=0

    3. The attempt at a solution

    I made somthing that for (a):
    [tex]R(right side)=V''+(2/r)V'[/tex]
    [tex]L(left side)={1 \over r}(rV)''={1 \over r}(r'V+rV')'={1 \over r}(r''V+r'V'+r'V'+rV'')=V''+(2/r)V'=R(right side)[/tex]

    And i have to use this equation to point (b):

    [tex] u''+{2 \over r}u'=1<=>{1 \over r}(ru)''=1[/tex]
    where: 1<r<2, u(1)=1, u(2)=2

    I know how to solve equation like this: [tex]u''+{2 \over r}u'=1[/tex], but I don't know how too solve [tex]{1 \over r}(ru)''=1[/tex]

    For second question:

    1. The problem statement, all variables and given/known data
    With witch k border problem:
    -u''-u=ku on (0, 1), u'(0)=1=u'(1)
    have solution?

    2. Relevant equations

    [tex] -u''+q(x)u=\lambda u[/tex]
    this is equation for Sturm-Liouville Border Problem

    3. The attempt at a solution
    I try something that: This can be rewritten as u''+(k+1)u=0
    but this is not this kind of solution. My teacher command me to use Sturm-Liouville Border Problem

    And third:

    1. The problem statement, all variables and given/known data
    Find restricted solution of border problem:

    [tex]\Delta u-u=|x|^2[/tex] where [tex]x \in [/tex]B1(0) tight in R^3
    u(x)=1 when |x|=1, |x| is euklides length of vector.
    Use spherical variables and find solution dependent on "r"

    2. Relevant equations
    [tex]\Delta={ 1 \over r^2} { \delta \over \delta r} r^2 { \delta \over \delta r} + {1 \over r^2 sin \Theta} { \delta \over \delta \Theta} sin \Theta { \delta \over \delta \Theta} + {1 \over r^2 sin^2 \Theta} { \delta^2 \over \delta \Gamma^2}[/tex]

    [tex]x=rsin \Theta cos \Gamma[/tex]
    [tex]y=rsin \Theta sin \Gamma[/tex]
    [tex]z=rcos \Theta[/tex]

    3. The attempt at a solution

    I made this in another way. I use Bessel potential and Fourier Transform, but I didn't use spherical variables
     
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