This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
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.
 Spring '09
 EWQFQW

Click to edit the document details