How to solve a basic newtons equation

If I have m((d^2s)/(dt^2))=F where F is constant and s(t_0)=s_0 s'(t_0)=v_0 how do I solve it using integration? not particular solution etc.

Sorry, I know that the notation is really bad, underscores are subscripts and ^ are powers. The s' is meant to be s dot. But I dont know how to put a dot over a letter.

Assuming a constant mass, m, integrate each side to get

m ds/dt = Ft + c, where c is constant

which is the same as

m s'(t) = Ft + c

Since s'(t_0)=v_0,

m s'(t_0) = F(t_0) + c
m v_0 = F(t_0) + c

so, c = mv_0 - F(t_0)

Therefore,

m s'(t) = Ft + mv_0 - F(t_0)

Integrate one more time to get

m s(t) = ½ Ft² + mv_0(t) - F(t_0)t + c1, where c1 is constant

Now use the given that s(t_0)=s_0 to solve for c1 and get the general solution.

If F is a constant, then integrating twice gives

s(t) = (F/m) t²/2 + A t + B

where A and B are constants of integration. Applying the initial conditions

s(t) = (F/m) (t - t_0)²/2 + v_0 (t - t_0) + s_0.

It is not at all necessary that F, v_0, and s_0 be scalars. They can just as easily be constant vectors, and this will still hold.