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?