Solving d^2x/dt^2 + a/x = b: Approximations?

[gabi.petrica]

I was trying to solve a physics problem which led me to an equation of the form:
d^2x/dt^2 + a/ x = b; Can this be solved without any aproximations beeing made?

 

[Matthew Rodman]

Depends.

Assuming a,b are constants, multiply by $$x^{\prime}$$, to get

$$x^{\prime} x^{\prime \prime} + a \frac{x^{\prime}}{x} = b x^{\prime}$$

now integrate to get

$$\frac{x^{\prime}^2}{2} + a \ln{x} = bx + c$$

(c is a constant). Rearrange to get t as a function of x, i.e.,

$$\frac{1}{\sqrt{2}} \int{\frac{dx}{\sqrt{bx + c - a \ln{x}}}} = t + const.$$

Other than that, I'm not sure.