Finding position and velocity from Force (analytic mechanics)

Homework Statement

A body of mass "m" is repelled from the origin by a force F(x). The body is at rest at x_0, a distance from the origin, at t=0. Find v(x) and x(t).

Homework Equations

$$F(x)=\frac{k}{x^3}$$
$$\ddot{x}=\frac{d\dot{x}}{dt}=\frac{dx}{dt}\frac{d\dot{x}}{dx}=v\frac{dv}{dx}$$

The Attempt at a Solution

I first perform the following steps:

$$F=ma=m\ddot{x}=\frac{k}{x^3}$$
$$\int{v{dv}}=\int_{x_0}^{x}\frac{k}{mx^3}dx$$
$$\frac{v^2}{2}=\frac{-k}{2mx^2}+\frac{k}{2m{x_0}^2}$$
$$v(x)=\sqrt{\frac{k}{m}(x_{0}^{-2}-x^{-2})}$$

The next step is to integrate v(x) WRT time. Doing this will solve for the position. My homework states that x(t) should be:

$$x(t)=\sqrt{\frac{kt^2}{mx_{0}^{2}}+x_{0}^{2}}$$

If I integrate v(x) WRT time, I will get a different answer for position than the equation above. Is my expression for v(x) incorrect?