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

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

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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 $s (t) = C 1 + \frac{20}{3} t^{3} + \frac{5}{2} t^{2} - 3 t$ .

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 $s (t) = - C 1 + \frac{20}{3} t^{3} + \frac{5}{2} t^{2} - 3 t + 116$ .

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

Find the particular solution determined by the initial condition.dsdt=20t2+5t−3;s(0)=116

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