Query 12 Differential Equations

Question:  A cylinder floating vertically in the water has radius 50 cm, height 100 cm and uniform density 700 kg / m^3.  It is raised 10 cm from its equilibrium position and released.  Set up and solve a differential which describes its position as a function of clock time.

The same cylinder, originally stationary in its equilibrium position, is struck from above, hard enough to cause its top to come just to but not below the level of the water.  Solve the differential equation for its motion with this condition, and use a particular solution to determine its velocity just after being struck.

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Question: 

For each of the equations and initial conditions below, find the largest t interval in which a solution is known to exist:

• y '' + y ' + 3 t y = tan(t), y(pi) = 1, y ' (pi) = -1

• t y '' + sin(2 t) / (t^2 - 9) y ' + 2 y = 0, y(1) = 0, y ' (1) = 1.

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Question:  Decide whether the solution of each of the following equations is increasing or decreasing, and whether each is concave up or concave down in the vicinity of the given initial point:

• y '' + y = - 2 t, y(0) = 1, y ' (0) = -1

• y '' + y = - 2 t, y(0) = 1, y ' (0) = -1

• y '' - y = t^2,   y(0) = 1, y ' (0) = 1

• y '' - y = - 2 cos(t), y (0) = 1, y ' (0) = 1.

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