classes_winter09_113AID28_Handout_3_Chem113A_W09

classes_winter09_113AID28_Handout_3_Chem113A_W09 - BRIEF...

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BRIEF REVIEW OF CONSTANT COEFFICIENT SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS This review follows ”Calculus” by Stewart, Edition 4, Chapter 18. First, we review some general facts about second-order linear differential equations. A second-order linear differential equation (SOLDE) has the form P ( x ) d 2 y dx 2 + Q ( x ) dy dx + R ( x ) y = G ( x ) (1) where P, Q, R and Q are continuous functions. If G ( x ) = 0 in (1), then P ( x ) d 2 y dx 2 + Q ( x ) dy dx + R ( x ) y = 0 (2) and the SOLDE is called homogeneous ; if G ( x ) 6 = 0, it is called nonhomo- geneous . FACT 1 : If y 1 ( x ) and y 2 ( x ) are solutions to the homogeneous SOLDE of (2), then so is any linear combination of y 1 and y 2 , i.e., y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) is also a solution to the homogeneous SOLDE in (2) for any constants c 1 and c 2 . Two functions y 1 ( x ) and y 2 ( x ) are called linearly independant if neither y 1 or y 2 is a constant multiple of the other, y 1 6 = cy 2 for c 6 = 0. FACT 2 : If y 1 ( x ) and y 2 ( x ) are linearly independant solutions to (2) and P ( x )
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