DiffEqProblems2(2).pdf

DiffEqProblems2(2).pdf - CHEN20041 Problem Sheet 2...

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CHEN20041 Problem Sheet 2: Second-Order Ordinary Differential Equations II 1. Find the general solution of d 2 y dx 2 + 2 dy dx - 15 y = 8 e 3 x . 2. Find the general solutions of (a) d 2 y dx 2 + 4 dy dx + 4 y = 12 e - 2 x (b) d 2 y dx 2 - 2 dy dx + 5 y = e 2 x (17 + 5 x ) 3. Find the general solution to d 2 y dx 2 + 4 dy dx - 12 y = f ( x ) where (a) f ( x ) = 8 e - 6 x (b) f ( x ) = - 24 e - 6 x + 72 x - 36 (c) f ( x ) = 8 e 2 x - 8 e - 6 x 4. An object (of unit mass) floating in water obeys the equation d 2 z dt 2 + k dz dt + az = 0 , where z ( t ) is the height of the object at time t , k governs a frictional force and a represents a restoring force (combination of gravity, buoyancy etc). In the case where a = 100, and starting from z (0) = 1, z 0 (0) = 0, find the solution to the ODE when (i) k = 12, (ii) k = 20,
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(iii) k = 25. If you have access to a graph plotting package, plot and compare the solu- tions. What happens if k = 0? 5. Consider an arrangement of two tanks A and B , where tank A is fed by a flow f ( t ), and tank A gives a flow of 2 y (where y is the current height of
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