m238sp05t2s - Differential Equations test two solutions...

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Unformatted text preview: Differential Equations test two solutions Monday, May 16, 2005 1 . For which values of t is the solution to the following differential equation guaranteed to exist? ( t 2- 9) y 00 + (cos t ) y + y = 2 t and y (0) =- 4 Rewrite the equation to isolate the highest derivative term: y 00 + cos t t 2- 9 y + 1 t 2- 9 y = 2 t t 2- 9 The coefficient functions are not continuous at t = 3. The existence & uniqueness theorem for second order linear equations states that a unique solution exists from the initial value of t = 0 up to values of t where the coefficient functions are discontinuous. So a solution will exist when- 3 < t < 3. 2 . Calculate the Wronskian of the functions f ( t ) = e 2 t and g ( t ) = e- t . W = f ( t ) g ( t ) f ( t ) g ( t ) = e 2 t e- t 2 e 2 t- e- t =- 3 e t 3 . Which of the following are linearly independent sets of functions? ( a ) { f ( t ) = t, g ( t ) = t 2 } ( b ) { f ( t ) = cos t, g ( t ) = t cos t } ( c ) { f ( t ) = e t- sin t, g ( t ) = sin t- e t } In (a) and (b), one function is not a constant multiple of the other, so each set is linearly independent. Things are not so wonderful in part (c), where the function g = (- 1) f , so that this set of functions is not linearly independent.that this set of functions is not linearly independent....
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This note was uploaded on 11/01/2010 for the course ENGINEERIN PHYS222 taught by Professor Ewqfqw during the Spring '09 term at University of Washington.

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m238sp05t2s - Differential Equations test two solutions...

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