Find the particular solution determined by the initial condition.
\[
\frac{d s}{d t}=20 t^{2}+5 t-3 ; s(0)=116
\]

Step 1 ：This is a first order ordinary differential equation. The general solution of this differential equation can be obtained by integrating the right hand side of the equation.

Step 2 ：After integrating, we get the general solution as \(s(t) = C1 + \frac{20}{3}t^3 + \frac{5}{2}t^2 - 3t\).

Step 3 ：We can use the initial condition \(s(0)=116\) to find the constant of integration and thus find the particular solution.

Step 4 ：Substituting the initial condition into the general solution, we get the particular solution as \(s(t) = -C1 + \frac{20}{3}t^3 + \frac{5}{2}t^2 - 3t + 116\).

Step 5 ：\(\boxed{s(t) = \frac{20}{3}t^3 + \frac{5}{2}t^2 - 3t + 116}\) is the particular solution of the differential equation.