Find the particular solution determined by the initial condition. \[ \frac{d s}{d t}=17 t^{2}+5 t-7 ; s(0)=86 \] The particular solution that satisfies the given condition is $s=$

Step 1 ：This is a first order ordinary differential equation. To find the particular solution, we need to integrate the right hand side of the equation and then apply the initial condition to solve for the constant of integration.

Step 2 ：Integrate the right hand side of the equation: \(\int (17 t^{2}+5 t-7) dt = \frac{17}{3} t^{3}+\frac{5}{2} t^{2}-7 t + C\)

Step 3 ：Apply the initial condition s(0)=86 to solve for the constant of integration C. Substituting t=0 and s=86 into the equation, we get C=86.

Step 4 ：Substitute C=86 back into the equation, we get the particular solution: \(s=\frac{17}{3} t^{3}+\frac{5}{2} t^{2}-7 t+86\)

Step 5 ：Final Answer: The particular solution that satisfies the given condition is \(\boxed{\frac{17}{3} t^{3}+\frac{5}{2} t^{2}-7 t+86}\)