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Homogeneous Second Order O.D.E Problem! Please help

  1. May 15, 2016 #1
    1. The problem statement, all variables and given/known data
    The electric potential energy v(r) of a charged particle located between two uniformly charged concentric spheres with radii r1 and r2 satisfies the second order differential equation

    rv′′+2v′=0, r1≤r≤r2
    where r is the distance of the charged particle from the common centre of the spheres.

    (a) Determine the general solution of the differential equation, by trying a solution of the form v(r)=rm, m∈R.

    (b) Using your answer to part (a), find the electric potential energy of a charged particle between two concentric spheres with radii r1 = 2 cm and r2 = 20 cm, kept at potentials v1 = 220 Volts and v2 = 130 Volts respectively.


    2. Relevant equations
    rv′′+2v′=0

    3. The attempt at a solution
    I tried to use v(r)=rm as a solution:
    v'(r)=mrm
    v''(r)=m2rm
    and sub back in:
    m2rm+1+2mrm=0
    this yields:
    m(mr+2)=0, which leads me no where near the solution

    Please help
     
  2. jcsd
  3. May 15, 2016 #2

    vela

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    You should rethink what v'(r) and v''(r) are equal to.
     
  4. May 15, 2016 #3
    Darn it I took the exponential. I'll rework my solution
     
  5. May 15, 2016 #4
    i found the general solution to be v(r)=A+Be^(-r). then, what's next?
     
  6. May 15, 2016 #5
    Apply these conditions:
    and you get A=200 and B=120
     
  7. May 15, 2016 #6
    is there any other way to simplify the exponential before applying the conditions?
     
  8. May 15, 2016 #7
    nawhh i just got the answer! thank you!
     
  9. May 15, 2016 #8
    wait howd u get the exponential? the solution i got was v=A/r+B
     
  10. May 15, 2016 #9
    yeah, i was wrong at first. no exponential. you are right :)
     
  11. May 15, 2016 #10
    isn't the general solution supposed to be v(r)=A+B/r? because both r^0 and r^-1 satisfy the DE
     
  12. May 15, 2016 #11
    oops you guys updated... yeah, and after getting v(r)=A+B/r, how do i sub in the conditions... the conditions doesn't make sense to me
     
  13. May 15, 2016 #12
    you will eventually get A=120 and B=200. just sub those conditions into v(r)=A+B/r separately and solve A and B simultaneously
     
  14. May 15, 2016 #13
    ohhhhh okay, i got the answer! even tho it doesn't make much sense... thank you for your help!
     
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