Homogeneous Second Order O.D.E Problem Please help

[i_hate_math]

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)=r^m as a solution:
v'(r)=mr^m
v''(r)=m²r^m
and sub back in:
m²r^m+1+2mr^m=0
this yields:
m(mr+2)=0, which leads me no where near the solution

Please help

 

[vela]

You should rethink what v'(r) and v''(r) are equal to.

 

[i_hate_math]

You should rethink what v'(r) and v''(r) are equal to.

Darn it I took the exponential. I'll rework my solution

 

[blerghh]

i found the general solution to be v(r)=A+Be^(-r). then, what's next?

 

[i_hate_math]

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

 

[blerghh]

Apply these conditions:

and you get A=200 and B=120

is there any other way to simplify the exponential before applying the conditions?

 

[blerghh]

nawhh i just got the answer! thank you!

 

[i_hate_math]

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

 

[blerghh]

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 :)

 

[crazy too]

isn't the general solution supposed to be v(r)=A+B/r? because both r^0 and r^-1 satisfy the DE

 

[crazy too]

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

 

[blerghh]

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

 

[crazy too]

ohhhhh okay, i got the answer! even tho it doesn't make much sense... thank you for your help!