wksht05sol - MATH 152 L1 & L2 Applied Linear Algebra...

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Unformatted text preview: MATH 152 L1 & L2 Applied Linear Algebra and Differential Equations Spring 2004-05 ¥ Week 05 Worksheet Solutions : Second-Order Linear Equations Q1. (The method of variation of parameters) The homogeneous equation ty 00- ( t + 1) y + y = 0 ( t > 0) has two linearly independent solutions y 1 ( t ) = t +1 and y 2 ( t ) = e t . By the method of variation of parameters, the nonhomogeneous equation ty 00- ( t + 1) y + y = t 2 , t > has solutions of the form u 1 ( t ) · ( t + 1) + u 2 ( t ) · e t for suitable choices of functions u 1 ( t ) and u 2 ( t ). (a) Assuming that u 1 ( t ) · ( t +1)+ u 2 ( t ) · e t = 0, derive the additional condition (another equation) that u 1 ( t ) and u 2 ( t ) must satisfy for u 1 ( t ) · ( t +1)+ u 2 ( t ) · e t to be a solution of the nonhomogeneous equation. Solution Since the nonhomogeneous equation has solutions of the form y ( t ) = u 1 ( t ) · ( t + 1) + u 2 ( t ) · e t , (1) then by differentiation, we have y ( t ) = u 1 ( t ) · ( t + 1) + u 2 ( t ) · e t + u 1 ( t ) + u 2 ( t ) · e t . Let u 1 ( t ) · ( t + 1) + u 2 ( t ) · e t = 0. Then we have y ( t ) = u 1 ( t ) + u 2 ( t ) · e t , (2) y 00 ( t ) = u 1 ( t ) + u 2 ( t ) · e t + u 2 ( t ) · e t . (3) Substituting (1)–(3) into the nonhomogeneous equation, we have t £ u...
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This note was uploaded on 09/30/2010 for the course MATH MATH152 taught by Professor Kcc during the Spring '10 term at HKUST.

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wksht05sol - MATH 152 L1 & L2 Applied Linear Algebra...

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