Review - solutions of ( t + 1) y 00 + t 2 y = 0. b)...

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Some review problems 1. a) Solve y 00 - 6 y 0 + 10 y = 0. b) Find the solution to a) satisfying y (0) = 1 , y 0 (0) = 2. In c)-h) below do not calculate the solution! c) Write the form of a particular solution to y 00 - 6 y 0 + 10 y = 3. d) Write the form of a particular solution to y 00 - 6 y 0 + 10 y = 3 t . e) Write the form of a particular solution to y 00 - 6 y 0 + 10 y = 3 e t . f) Write the form of a particular solution to y 00 - 6 y 0 + 10 y = 3 te t . g) Write the form of a particular solution to y 00 - 6 y 0 + 10 y = 3 cos(3 t ). 2. For the problems c)-h) above: a) Calculate a particular solution to the problems , then check your answer by substituting in the equation. b) Write the general solution. c) Find the solution satisfying y (0) = 0 , y 0 (0) = 1. 3. a) Solve y 00 - 3 y 0 + 2 y = 0. b) Solve y 00 - 3 y 0 + 2 y = 2 t by way of finding a particular solution. c) Solve b) using the variation of parameters. 4. a) Find the differential equation satisfied by the Wronskian of two independent
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Unformatted text preview: solutions of ( t + 1) y 00 + t 2 y = 0. b) Calculate the Wronskian of two solutions y 1 , y 2 of the equation in a) which satisfy y 1 (0) = 1 , y 1 (0) = 0, respectively y 2 (0) = 0 , y 2 (0) = 1. c) Are the solutions in b) linearly independent? 5. a) Solve t 2 y 00 + ty-y = 0. b) Solve t 2 y 00 + ty-y = 1. c) Solve t 2 y 00 + ty-y = 1 /t . (Use the variation of parameters.) 6. Consider the differential equation x 2 (2 x + 1) d 2 dx 2 y ( x )-x (4 + 7 x ) d dx y ( x ) + 2 (5 x + 3) y ( x ) a) Show that y ( x ) = x 2 is a solution of the equation. b) Find the general solution of the equation. 1...
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This note was uploaded on 07/17/2008 for the course MATH 415 taught by Professor Costin during the Fall '07 term at Ohio State.

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