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15] [a5] {15} Math 246H Final Exam Fall 2006 M. Jakobson Solve each of the following eight problems on a separate answer sheet, using the back of the sheet if necessary. Put your name, section number, the name of your recitation instructor, and th number of the problem on each sheet. Carefully show all your steps and explain your arguments. Cross out any material that is not to be graded. You may NOT use calculators, notes ‘or texts. 1, Solve the initial value problem| Find the maximal interval in which the solution is valid. at 5) =2 Y* aoe y(5} 2. (a) A tank contains 10 kg of galt dissolved in 100 liters of water. A solution of salt water containing 1 kg/liter flows into the tank at a rate of 5 liters/minute, tank at the same rate. and well mixed solution flows out of the i, Write down an initial yalue problem satisfied by the amount of salt Q(t) in the tank at time ¢. ‘ ii. Solve the initial value problem. iii. Determine the limiting amount of salt in the tank when to (b) For the differential equatioi N= (9-N*)2—N) i, Determine equilibrium |solutions and classify them from the point of view of stability. ii. Sketch the solution curves as functions of ¢. tii. Let Ni(t) be the solution with the initial condition NO) = 1.1 and let No(t) be N(0) = 2.8. Find the solution with the initial condition dap No(t) — y(t) bo 3. (a) A certain vibrating system satisfies the equation a u +/2yu' +u=0 i. Find the critical damping +,,. ii. Sketch sample solutions in three cases Y< Yer: = Yer , and Y> Yer 
