Diff. eq. for the Einstein metric inside a body

[gijeqkeij]

Solving the Einstein eq. i found the following differential eq. that would descrive the metric inside a body:
x2d2y(x)/dx2+2xD1(y)dy(x)/dx+D2(y)=0
where D1(y) and D2(y) are known function of y and 0<=x<=1.
I try to solve numerically but looks like there is a cusp; any suggestion how to approach that eq. or some links where that eq. is treated?
Ty

[gijeqkeij]

If we change the x with x=exp(-t) we have the following diff. eq.:
d2y/dt2+(1-2D1)dy/dt+D2=0
where 0<=t<=\infty.

Any hint? Do you know where this kind of eq. has been analyzed?

Ty

[MrSnoopy]

Hi.

Yes this is very difficult task you want to solve, but for my opinion is unsolvable in analytical way!
Why?
You don't know what kind of functions are D_{1}(y), D_{2}(y) this is main problem and even not knowing whether you can solve analytical or numerical, but for solving numerical this is very difficult task. If you know what function are D_{1}(y), D_{2}(y) than is trivial task as you already know :smile:

[gijeqkeij]

Hi.

Yes this is very difficult task you want to solve, but for my opinion is unsolvable in analytical way!
Why?
You don't know what kind of functions are D_{1}(y), D_{2}(y) this is main problem and even not knowing whether you can solve analytical or numerical, but for solving numerical this is very difficult task. If you know what function are D_{1}(y), D_{2}(y) than is trivial task as you already know :smile:

Thanks Snoopy. Actualli I know the D1(y)=A1*(B1-ye)/ye+1/2 and D2(y)=A2(1-ye)(B2-ye)/y2e-1... (Ai, Bi, e are costants) but the problems is that when i try to solve that equation numerically y->0 for a finite value of t and that has no physical meaning (the metric degenerates at a finite (>0) distance from the center of the star). Why u said it's trivial if i would know D1 and D2?

[MrSnoopy]

No problem gijeqkeij.
It is only expression we use to simplify our life :smile:

Trivial task is only when"thinking" in numerical solving problem without analytical solution.
On the other hand you can see, with your note, that D(y), y \neq y(x) but you are want to solve DE y''(t)+C_{1}\; y'(t)= C_{2}
where C_{1}, C_{2} are only constants.
If I understand you correctly D_{1,2}(y),\; D_{1,2} \neq D_{1,2}(y(t)) , if so you just have to solve DE second degree analytical :wink:

Glad to help
MrSnoopy

[gijeqkeij]

No problem gijeqkeij.
It is only expression we use to simplify our life :smile:

Trivial task is only when"thinking" in numerical solving problem without analytical solution.
On the other hand you can see, with your note, that D(y), y \neq y(x) but you are want to solve DE y''(t)+C_{1}\; y'(t)= C_{2}
where C_{1}, C_{2} are only constants.
If I understand you correctly D_{1,2}(y),\; D_{1,2} \neq D_{1,2}(y(t)) , if so you just have to solve DE second degree analytical :wink:

Glad to help
MrSnoopy

Sorry Snoopy I guess I was unclear. The eq. to solve is: y''(t)+C_{1}(y(t))\; y'(t)= C_{2}(y(t)). I thought that kind of eq. received some attention in literature and I wonder if somebody can help me in finding papers that analyze that type of DE. Thanks

[MrSnoopy]

Hy :smile:

You wrote something like this:
D_{1}(y) = A_{1}\frac{B_{1}-y^{e}}{y^{e}}+\frac{1}{2} and
D_{2}(y) = A_{2} (1-y^{e})\frac{B_{2}-y^{e}}{y^{2e-1}}\;\;\; ...

where are A_{1}, A_{2}, B_{1},B_{2}, e=exp(1) \in const. OK this DE cannot be solve analytical :smile:

Numerical approach :wink:

x^{2}y''(x)+\left[2xA_{1}(B_{1}y(x)^{-e}-1)+x \right]y'(x) +A_{2}(B_{2}-y^{e})(y^{1-e}-y^{1-2e}) = 0

For knowing y(x=?)= ..., y'(x= ?)= ... than make DE something like this:
y'_{1} = y_{2}
y'_{2} = -\frac{1}{x^{2}}\left[\left(2A_{1}x(B_{1}y^{-e}_{1}-1)+x\right)y_{2}+A_{2}(B_{2}-y^{e}_{1})(y^{1-e}-y^{1-2e})\right]

This is best to solve with Runge-Kutta 5 degree or with ode45 in MatLab or Octave.
I attach file with Runge-Kutta method in file.m (MatLab, Octave).
Write like this in MatLab:
f=inline([\; 'y(1)' \; ;\; '\frac{1}{x^2}\left[..... \right]'\;], 'x' , 'y' )

In Runge-Kutta file will output vector dimension 2*n where n is steps make in calculation.
Where you want do solve this equation on interval I second column in function RungeKutta5, y0(x=?)=[y(x=?) , y'(x=?)] and for n put 100 or more.

I hope I helped you :smile:

MrSnoopy

[gijeqkeij]

Thanks Snoopy... that help but the problem is that a numerical solution bring to a singularity somewhere for t=t* (finite value). I wonder if somebody has analyzed that kind of eq. to identify what are (if any) the conditions for it's integrability.