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Find the function s(t) satisfying ds/dt= -5- 3 cost and s(0) = 6.

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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.
User Shivansh Goel
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