Lecture 19

Lecture 19 - Section 4.6, contd, Section 4.7 Variation of...

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Unformatted text preview: Section 4.6, contd, Section 4.7 Variation of Parameters/Variable Coefficients April 8, 2009 Variation of Parameters/Variable Coefficients Todays Session A Summary of This Session: (1) Justification of Variation of Parameters. (2) Variable-Coefficient Equations (Eulers Method). Variation of Parameters/Variable Coefficients Variation of Parameters Suppose you want to find a particular solution to the constant coefficient differential equation ay + by + cy = f ( t ) Find the fundamental solutions of the homogeneous equation y 1 and y 2 . Find a particular solution of the form y p = v 1 ( t ) y 1 ( t ) + v 2 ( t ) y 2 ( t ) by solving the system braceleftBig v 1 y 1 + v 2 y 2 = v 1 y 1 + v 2 y 2 = f ( t ) a Variation of Parameters/Variable Coefficients Variation of Parameters The answers are given by: v 1 =- f ( t ) y 2 a W ( y 1 , y 2 ) and v 2 = f ( t ) y 1 a W ( y 1 , y 2 ) Or v 1 =- integraldisplay f ( t ) y 2 a W ( y 1 , y 2 ) and v 2 = integraldisplay...
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Lecture 19 - Section 4.6, contd, Section 4.7 Variation of...

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