When you use four-velocities, four-accelerations, and four-forces, the laws become much simpler:
If you have a point mass m, with cartesian coordinates t,x,y,z and ##\tau## representing proper time, and you parameterize all of [t,x,y,z] as functions of ##\tau##, i.e you write ##t(\tau), x(\tau), y(\tau), z(\tau)##
Then the 4-velocity is a vector with 4 componenets: ##[u^0, u^1, u^2, u^3]##
where ##u^0 = \frac{dt}{d\tau}##, ##u^1 = \frac{dx}{d\tau}##, ##u^2 = \frac{dy}{d\tau}##, ##u^3 = \frac{dz}{d\tau}##
if you let ##\chi^0 = t##, ##\chi^1 = x##, ##\chi^2 = y##, ##\chi^3 = z## this can be compactly written as
u^\mu = \frac{d\chi^\mu}{d\tau}
Continuing with the compact notation, then
The 4-acceleration is ##a^\mu = \frac{d}{d\tau} u^\mu = \frac{d^2 \chi^\mu}{d \tau^2}##
Then the 4-momentum is ##p^\mu = m \, u^\mu##
the four-force is ##f^\mu = \frac{d p^\mu}{d\tau} = m a^u##
Knowing the time dilation factor, you can find the components of the three-fource from the four-force as needed, knowing that the 3-force is ##\frac{dp}{dt}## while the 4-fource is ##\frac{dp}{d\tau}##
