• Support PF! Buy your school textbooks, materials and every day products Here!

Help me with this

  • Thread starter ywel84
  • Start date
  • #1
2
0
Hello!!!
My English is poor because I'm from Poland.

I have 3 exercises from subject called: Mathematical Method In Physics.
Would you be so kind and help me to solve this:

1.
(a) Prove that: [tex]{1 \over r}(rV)''=V''+(2/r)V' [/tex](that was easy, but next point (b) is not)
(b) Use equation from (a) to solve:
[tex] u''+{2 \over r}u'=1[/tex], where 1<r<2, u(1)=1, u(2)=2

2.
With witch k border problem:
-u''-u=ku on (0, 1), u'(0)=1=u'(1)
have solution?

3.
Find restricted solution of border problem:

[tex]\Delta u-u=|x|^2[/tex] where [tex]x \in [/tex]B1(0) tight in R^3
u(x)=1 when |x|=1, |x| is euklides length of vector.
Use spherical variables and find solution dependent on "r"

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
berkeman
Mentor
56,841
6,823
Welcome to the PF, ywel84. One of the rules here is that you must show your own work so far, before we can help you out. We do not provide answers here, but we can provide helpful hints and so forth, after you fill in your answers for questions #2 and #3 in the Homework Help Template above.
 
  • #3
cristo
Staff Emeritus
Science Advisor
8,107
73
Hello!!!
My English is poor because I'm from Poland.

I have 3 exercises from subject called: Mathematical Method In Physics.
Would you be so kind and help me to solve this:

1.
(a) Prove that: [tex]{1 \over r}(rV)''=V''+(2/r)V' [/tex](that was easy, but next point (b) is not)
(b) Use equation from (a) to solve:
[tex] u''+{2 \over r}u'=1[/tex], where 1<r<2, u(1)=1, u(2)=2
What have you tried? Have you used part (a)?

2.
With witch k border problem:
-u''-u=ku on (0, 1), u'(0)=1=u'(1)
have solution?
Hint: This can be rewritten as u''+(k+1)u=0
3.
Find restricted solution of border problem:

[tex]\Delta u-u=|x|^2[/tex] where [tex]x \in [/tex]B1(0) tight in R^3
u(x)=1 when |x|=1, |x| is euklides length of vector.
Use spherical variables and find solution dependent on "r"
Again, what have you tried?

Please note that for homework questions, we must see your work before we can help. Also, in future, please post in the homework forums!

edit: damn, beaten to it!
 
  • #4
2
0
For first exercise:

Homework Statement


(a) Prove that: [tex]{1 \over r}(rV)''=V''+(2/r)V' [/tex]
(b) Use equation from (a) to solve:
[tex] u''+{2 \over r}u'=1[/tex], where 1<r<2, u(1)=1, u(2)=2

Homework Equations

//I don't understand this
r'=1
r''=0

The Attempt at a Solution



I made somthing that for (a):
[tex]R(right side)=V''+(2/r)V'[/tex]
[tex]L(left side)={1 \over r}(rV)''={1 \over r}(r'V+rV')'={1 \over r}(r''V+r'V'+r'V'+rV'')=V''+(2/r)V'=R(right side)[/tex]

And i have to use this equation to point (b):

[tex] u''+{2 \over r}u'=1<=>{1 \over r}(ru)''=1[/tex]
where: 1<r<2, u(1)=1, u(2)=2

I know how to solve equation like this: [tex]u''+{2 \over r}u'=1[/tex], but I don't know how too solve [tex]{1 \over r}(ru)''=1[/tex]

For second question:

Homework Statement


With witch k border problem:
-u''-u=ku on (0, 1), u'(0)=1=u'(1)
have solution?

Homework Equations



[tex] -u''+q(x)u=\lambda u[/tex]
this is equation for Sturm-Liouville Border Problem

The Attempt at a Solution


I try something that: This can be rewritten as u''+(k+1)u=0
but this is not this kind of solution. My teacher command me to use Sturm-Liouville Border Problem

And third:

Homework Statement


Find restricted solution of border problem:

[tex]\Delta u-u=|x|^2[/tex] where [tex]x \in [/tex]B1(0) tight in R^3
u(x)=1 when |x|=1, |x| is euklides length of vector.
Use spherical variables and find solution dependent on "r"

Homework Equations


[tex]\Delta={ 1 \over r^2} { \delta \over \delta r} r^2 { \delta \over \delta r} + {1 \over r^2 sin \Theta} { \delta \over \delta \Theta} sin \Theta { \delta \over \delta \Theta} + {1 \over r^2 sin^2 \Theta} { \delta^2 \over \delta \Gamma^2}[/tex]

[tex]x=rsin \Theta cos \Gamma[/tex]
[tex]y=rsin \Theta sin \Gamma[/tex]
[tex]z=rcos \Theta[/tex]

The Attempt at a Solution



I made this in another way. I use Bessel potential and Fourier Transform, but I didn't use spherical variables
 

Related Threads on Help me with this

  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
1
Views
890
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
1
Views
4K
Replies
0
Views
887
Replies
5
Views
7K
Replies
3
Views
1K
Replies
7
Views
1K
Replies
2
Views
539
Top