Solving Mathematical Method In Physics Exercises

I need too use these variables and find solution dependent on "r".In summary, the conversation discusses three exercises in the subject of Mathematical Methods in Physics. The first exercise involves proving an equation and using it to solve another equation. The second exercise involves finding a solution for a Sturm-Liouville Border Problem. The third exercise involves finding a restricted solution using spherical variables and finding a solution dependent on "r". The person asking for help has provided some of their attempts at solving the exercises, but is seeking further assistance.
  • #1
2
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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"
 
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  • #2
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
ywel84 said:
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
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
 

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The purpose of solving mathematical method in physics exercises is to develop critical thinking skills and problem-solving abilities. These exercises allow scientists to apply mathematical concepts to real-world situations and test their understanding of physical principles.

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To check if your solution to a mathematical method in physics exercise is correct, you can use various methods such as plugging in the calculated values into the original equation to see if they match, double-checking your calculations, or comparing your solution with a known correct answer.

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