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

  • #1
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Homework Statement


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.


Homework Equations


rv′′+2v′=0

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
 

Answers and Replies

  • #2
vela
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You should rethink what v'(r) and v''(r) are equal to.
 
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  • #3
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You should rethink what v'(r) and v''(r) are equal to.
Darn it I took the exponential. I'll rework my solution
 
  • #4
5
1
i found the general solution to be v(r)=A+Be^(-r). then, what's next?
 
  • #5
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i found the general solution to be v(r)=A+Be^(-r). then, what's next?
Apply these conditions:
two concentric spheres with radii r1 = 2 cm and r2 = 20 cm, kept at potentials v1 = 220 Volts and v2 = 130 Volts respectively.
and you get A=200 and B=120
 
  • #6
5
1
Apply these conditions:

and you get A=200 and B=120
is there any other way to simplify the exponential before applying the conditions?
 
  • #7
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nawhh i just got the answer! thank you!
 
  • #8
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Apply these conditions:

and you get A=200 and B=120
wait howd u get the exponential? the solution i got was v=A/r+B
 
  • #9
5
1
wait howd u get the exponential? the solution i got was v=A/r+B
yeah, i was wrong at first. no exponential. you are right :)
 
  • #10
3
1
isn't the general solution supposed to be v(r)=A+B/r? because both r^0 and r^-1 satisfy the DE
 
  • #11
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1
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
 
  • #12
5
1
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
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
 
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  • #13
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ohhhhh okay, i got the answer! even tho it doesn't make much sense... thank you for your help!
 
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