There exists a particle of mass $m$ acted on by a force $F = kt^2$. $k$ is a constant and $t$ is time. The particle starts at $x=0$ with constant velocity $u$. Find the acceleration, velocity and position of the particle.
Attempt:
$kt^2 = m\ddot x$, so $\ddot x = \frac{kt^2}{m}$ is the acceleration.
Integrating and using initial conditions $\dot x(0)=u$ and $x(0)=0$ gives $\dot x = \frac{kt^3}{3m} + u$ and $x = \frac{kt^4}{12m}+ut$.
Is this correct ?