M427Kfins10a - linearly independent solution and then show...

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M427K Final Exam A Name NO NOTES. NO CALCULATORS. 1. Solve the IVP y 0 + 2 y = te - 2 t , y (1) = 0 . 2. Find the first 3 non-zero terms of two linearly independent series solutions to (1 - x ) y 00 + y = 0 about x 0 = 0 . 3. Replace the PDE 5 tu xx + xu t by a pair of ordinary DE’s. 4. Find the Fourier series for the function f ( x ) = x + 2 if - 2 x < 0 2 - x if 0 x < 2 and f ( x + 4) = f ( x ). 5. Use Laplace transforms to solve y 00 + 4 y = 1 + 2 δ ( t - 5) , y (0) = y 0 (0) = 0 . 6. Find the general solution of y 00 + 9 y = 9 sec 2 3 t. 7. Note that y 1 = t is a solution to t 2 y 00 + 2 ty 0 - 2 y = 0. Use reduction of order to find a second,
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Unformatted text preview: linearly independent solution and then show that it is linearly independent. 8. A 4-foot long rod is made from a mysterious substance from Uranus, which has thermal diffusivity constant α 2 = 2. The initial temperature in the rod is given by the function in Problem 4. The left end of the rod is held at 0 degrees C while the right end is insulated. Write the BVP whose solution u ( x,t ) gives the temperature in the rod at point x and time t ....
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