M427Kfinf10a - the function which gives the amount of salt...

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M427K Final Exam A Name NO NOTES. NO CALCULATORS. 1. Find the general solution to (2 e 2 x cos y + 6 x ) dx - (1 + e 2 x sin y ) dy = 0 . 2. Use the method of undetermined coefficients to find the general solution to y 00 - 2 y 0 = 4 t . 3. Find the first 4 nonzero terms of the series solution about x = 0 to the IVP ( x - 1) y 00 + xy 0 - 5 y = 0 , y (0) = 2 , y 0 (0) = 3 . 4. A large tank initially contains 30 gallons of water and 5 pounds of salt. A salt solution consisting of 1 lb/gal is pumped into the tank at a rate of 3 gal/min, and the well-mixed solution is pumped out at the same rate. At time t = 10 an large hopper begins dumping salt into the tank at a rate of 5 lbs/min. At time t = 20, the hopper starts making an alarming noise, and the flow of salt stops. Then at time t = 30, the hopper ruptures and dumps 10 lbs of salt into the tank all at once. Find
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Unformatted text preview: the function which gives the amount of salt in the tank at time t . 5. Use the eigenvalue/egenvector method to nd the general solution to the system x 1 = 3 x 1-4 x 2 x 2 =-2 x 1 + 5 x 2 . 6. A 3-foot long rod is made from a mysterious substance from Uranus, which has thermal diusivity constant 2 = 2. The initial temperature in the rod is given by 4 x for 0 x 3. Both ends of the rod are held at 0 degrees C. Write the BVP whose solution u ( x,t ) gives the temperature in the rod at point x and time t , and then solve it. (Give the rst 4 nonzero terms of the solution.)...
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