Solving Mathematical Method In Physics Exercises

[ywel84]

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

$$\Delta u-u=|x|^2$$ where $$x \in$$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"

 

[berkeman]

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.

 

[cristo]

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

$$\Delta u-u=|x|^2$$ where $$x \in$$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!

 

[ywel84]

For first exercise:

Homework Statement

(a) Prove that: $${1 \over r}(rV)''=V''+(2/r)V'$$
(b) Use equation from (a) to solve:
$$u''+{2 \over r}u'=1$$, 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):
$$R(right side)=V''+(2/r)V'$$
$$L(left side)={1 \over r}(rV)''={1 \over r}(r'V+rV')'={1 \over r}(r''V+r'V'+r'V'+rV'')=V''+(2/r)V'=R(right side)$$

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

$$u''+{2 \over r}u'=1<=>{1 \over r}(ru)''=1$$
where: 1<r<2, u(1)=1, u(2)=2

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

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

$$-u''+q(x)u=\lambda u$$
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:

$$\Delta u-u=|x|^2$$ where $$x \in$$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

$$\Delta={ 1 \over r^2} { \delta \over \delta r} r^2 { \delta \over \delta r} + {1 \over r^2 sin \Theta} { \delta \over \delta \Theta} sin \Theta { \delta \over \delta \Theta} + {1 \over r^2 sin^2 \Theta} { \delta^2 \over \delta \Gamma^2}$$

$$x=rsin \Theta cos \Gamma$$
$$y=rsin \Theta sin \Gamma$$
$$z=rcos \Theta$$

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