To find the function s(t) that satisfies the given differential equation ds/dt = -5 - 3cos(t) with the initial condition s(0) = 6, we can integrate both sides of the equation with respect to t.
∫ ds = ∫ (-5 - 3cos(t)) dt
Integrating the right side, we have:
s(t) = -5t - 3∫cos(t) dt
The integral of cos(t) is sin(t), so we can continue:
s(t) = -5t - 3sin(t) + C
To determine the constant of integration (C), we'll use the initial condition s(0) = 6:
6 = -5(0) - 3sin(0) + C
Since sin(0) = 0, this simplifies to:
6 = 0 + C
Therefore, C = 6. Plugging this back into the equation, we have:
s(t) = -5t - 3sin(t) + 6
So, the function s(t) satisfying the given conditions is s(t) = -5t - 3sin(t) + 6.