1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Linear operator on the set of polynomials

  1. Jun 7, 2010 #1
    1. The problem statement, all variables and given/known data

    Let L be the operator on P_3(x) defined by

    L(p(x)) = xp'(x)+p"(x)

    if p(x) = a_0(x)+a_1(x)+a_2(1+x^2) calculate L^n(p(x))


    2. Relevant equations

    stuck between 2 possible solutions

    i) as powers of x decrease the derivatives of p(x) increase

    ii) as derivatives of x decrease the derivatives of p(x) increase

    3. The attempt at a solution

    check with increasing powers


    L^2(p(x)) = x^2p''(x)+xp"'(x)+p^(4)(x)

    L^3(p(x)) = x^3p'''(x) + x^2p^(4)(x)+xp^(5)(x)

    => L^n(p(x)) = x^np^(n)(x)+x^(n-1)p^(n+1)(x)+x^(n-2)p^(n+2)(x)

    when a superscript is applied to an x it is a power

    when a superscript is applied to a p it is a derivative

    I'm not quite sure which it should be

    any help is appreciated, thank you
     
  2. jcsd
  3. Jun 7, 2010 #2

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    What does
    p(x) = a_0(x)+a_1(x)+a_2(1+x^2) ​
    mean?

    Are a_0, a_1 and a_2 numbers, multiplied by x? So basically,
    [tex]p(x) = (a_0 + a_1) \cdot x + a_2 \cdot (1 + x^2) ?[/tex]
    Or are they functions of x and 1 + x2?

    Note that, if p lies in P3, then p(k) = 0 for all k > 3.
     
  4. Jun 7, 2010 #3
    the a's are coefiicients multiplied by the x's, the first x is x to the power 0, the second is x to the power 1 and then third is (1+x^2)

    so its a_0 by itelf + a_1 multiplied by x and a_2 multiplied by (1+x^2)

    sorry for the confusion
     
  5. Jun 7, 2010 #4

    Mark44

    Staff: Mentor

    Is there some reason p(x) is not a_0 + a_1*x + a_2 * x^2? That would seem more natural to me.
     
  6. Jun 7, 2010 #5

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    OK, that makes it more clear.
    Maybe it helps if you write out the first few lines explicitly.
    For example,

    [tex]L(p) = a_1 x + 2 a_2(1 + x^2) [/tex]
    [tex]L(L(p)) = \cdots ?[/tex]

    The pattern is quite hard to miss.
     
  7. Jun 7, 2010 #6

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    I think the basis is a given in the problem because the matrix for L has a nice form in that basis.
     
  8. Jun 8, 2010 #7

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    p is a polynomial of degree 3 or less. The fourth derivative of such a polynomial is 0!

     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook