Pts consider the two interconnected tanks let x 1 t

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8. (5 pts.) Consider the two interconnected tanks. Let x 1 ( t ) and x 2 ( t ) be the amount of salt in Tank1 and Tank2 at time t , respectively. Find the matrix A to represent the system by the differential equation x 0 1 ( t ) x 0 2 ( t ) = A x 1 ( t ) x 2 ( t ) , if both tanks initially contained 30 liters of salt water. (Note that for each tank the total inflow rate is the same as outflow rate, so that the volume of the salt water in each tank remains the same at all times.) (a) A = - 1 1 1 - 1 (b) A = 1 / 6 1 / 15 1 / 6 1 / 6 (c) A = - 1 / 6 1 / 10 1 / 6 - 1 / 6 (d) A = - 1 / 6 1 / 15 1 / 6 - 1 / 6 (e) none of the above
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9. (12 pts.) A mass of 20 g stretches a spring 5 cm. Suppose that the mass is also attached to a viscous damper with a damping constant of 400 g/s. If the mass is pulled down an additional 2 cm and then released, set up the initial value problem for u ( t ) at any time t 0. Do NOT solve the differential equation. Note: Don’t forget the initial conditions! 10. (12 pts.) Solve the initial value problem: y 0 - 4 t y = t 4 cos t , y ( π ) = π 5 ; t > 0. 4

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