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Buoyancy : Equilibrium Position of a Floating Cylinder

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I am trying to work through some example problems from Schaum's Outlines: Differential Equations.

I'm not sure how to come up with the solution and need help.

Based on the attached file the question is:
Find an expression for the motion of the cylinder described in previous problem if it is disturbed from its equilibrium position by submerging an additional 2 inches of height below the water line and with a velocity of 1 ft/sec in the downward direction.

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The expert examines equilibrium position of a floating cylinder for buoyancy.

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