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From weinberg's gravitation, EQ : $3.2.11$

$$\frac{\partial^2 \zeta^\alpha}{\partial x^\mu \partial x^\nu} = \Gamma^\lambda _{\mu \nu}\frac{\partial\zeta^\alpha}{\partial x ^\lambda}$$

The solution to this differential equation is given as

$$\zeta^\alpha(x) = a^\alpha + b^\alpha _{\mu}(x^\mu - X^\mu) + \frac{1}{2} b^\alpha _\lambda\Gamma^\lambda _{\mu \nu}(x^\mu - X^\mu)(x^\nu - X^\nu) + \cdot \cdot \cdot $$

How can it be solved?

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Should that be $\partial x^\mu$ rather than $\partial x^\lambda$ on the left? –  Chris White Jan 16 '13 at 8:53
Thanks for the noticing that. I've corrected it –  Aftnix Jan 16 '13 at 9:02
I'm having trouble with the language here. Paraphrasing: "The solution to this differential equation is ...", then: "How can it be solved?" Do you mean to ask: "How does one show that this solution (2nd equation) satisfies the equation above (1st equation)?"? –  MarkWayne Jan 16 '13 at 21:23
@MarkWayne yes i'm saying " how i can obtain the solution" –  Aftnix Jan 18 '13 at 16:38
@Aftnix: I think all that's going on here is plugging into the equation the general Taylor expansion (up to second order terms) and comparing coefficients. Do you need an explanation how to do this? –  Marek Jan 29 '13 at 12:36
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