1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Reduction of order of an ODE

  1. Feb 22, 2013 #1
    1. The problem statement, all variables and given/known data

    Use the method of reduction of order to find another independently linear solution y2(x) when given one solution.

    [tex] x^2y'' - x(x+2)y' + (x+2)y = 0 [/tex]

    [tex] y_1(x) = x [/tex]

    3. The attempt at a solution

    Hopefully y2(x) will take the form of v(x)y1(x) or I have no idea how to solve the ODE. I start by finding y'2(x) and y''2(x).

    [tex] y_2 = vx [/tex]
    [tex] y'_2= v + xv' [/tex]
    [tex] y''_2 = v' + v' + xv'' = 2v' + xv'' [/tex]

    I substitute the above into the equation:

    [tex] x^2(2v' + xv'') - x(x+2)(v + xv') + (x+2)vx = 0 [/tex]

    [tex] 2x^2v' + x^3v'' - x^2v - x^3v' - 2xv - 2x^2v' + x^2v + 2xv = 0 [/tex]

    And this is where I've gotten to. Everything cancels out and I can't see how I'll find my v...
  2. jcsd
  3. Feb 22, 2013 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    You have two terms left over: ##x^3 (v'' - v')=0##.
  4. Feb 22, 2013 #3
    Oh, wow. I overlooked the same thing three times...

    Might be worth switching notation to avoid this from happening again.

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook