Solve the initial value problem. \[ \frac{d s}{d t}=20 t\left(5 t^{2}-3\right)^{3}, \quad s(1)=5 \]

Step 1 ：The given differential equation is \(\frac{d s}{d t}=20 t\left(5 t^{2}-3\right)^{3}\) with the initial condition s(1) = 5.

Step 2 ：We start by integrating both sides of the equation. The integral of the right hand side, \(20t(5t^2 - 3)^3 dt\), can be solved using the power rule for integration.

Step 3 ：The general solution of the differential equation is \(s(t) = C1 + \frac{625}{2}t^8 - 750t^6 + 675t^4 - 270t^2\).

Step 4 ：We then substitute the initial condition, t = 1, into the general solution to find the constant of integration, C1. This gives us \(s(1) = C1 - \frac{65}{2}\).

Step 5 ：Solving for C1, we find that the constant of integration is \(C1 = \frac{75}{2}\).

Step 6 ：Substituting C1 back into the general solution, we obtain the final solution to the initial value problem: \(s(t) = \frac{625}{2}t^8 - 750t^6 + 675t^4 - 270t^2 + \frac{75}{2}\).

Step 7 ：\(\boxed{s(t) = \frac{625}{2}t^8 - 750t^6 + 675t^4 - 270t^2 + \frac{75}{2}}\) is the final solution to the initial value problem.