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Lecture 18 - Section 4.6 Variation of Parameters Variation...

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Unformatted text preview: Section 4.6 Variation of Parameters April 07, 2009 Variation of Parameters Today’s Session The average for the exam was 68. The median was 76. The TAs will go through the exam tomorrow, and there will be a quiz. I will discuss the hand-in today in class. A Summary of This Session: (1) Wronskian of independent solutions. (2) Variation of parameters to find particular solution, when the forcing function is not an, exponential, a polynomial, sine, cosine, or a combination of those. Variation of Parameters Wronskian of two solutions The Wronskian of two functions y 1 and y 2 is defined by: W ( y 1 , y 2 ) = vextendsingle vextendsingle vextendsingle y 1 y 2 y ′ 1 y ′ 2 vextendsingle vextendsingle vextendsingle = y 1 y ′ 2 − y 2 y ′ 1 Example: Find the Wronskians of the following: (a) cos t , sin t (b) t 2 + 2 t − 1 , t − 2 (c) e 2 t , t e 2 t Answer: (a) W ( y 1 , y 2 ) = 1; (b) W ( y 1 , y 2 ) = − t 2 + 4 t + 3; (c) W ( y 1 , y 2 ) = e 4 t . Variation of Parameters Wronskian, cont’d Example 2: (a) Show that the wronskian of the fundamental solutions of the differential equation y ′′ − 6 y ′ + 9 y = 0 is not zero....
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