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Unformatted text preview: 3.1: Introduction: Second-Order Linear Equations Recall: A differential equation is l i n e a r if it is linear in its dependent variables and derivatives. Thus x 2 y '' + y = 3is linear, however x 2 H y '' L 2 + y = 3is not linear. Definition A second order linear differential equation can be written in the form A H x L y '' + B H X L y ' + C H x L y = F H x L where A, B, C, F are continuous on an I . If F H x L 0 for all x in I , then we say it is h o m o g e n e o u s . Otherwise, it is n o n- h o m o g e n e o u s . Theorem - Principle of Supperposition for Homogeneous Equations Let y 1 and y 2 be two solutions to y '' + p H x L y ' + q H x L y = 0. Then y = c 1 y 1 + c 2 y 2 is also a solution. Theorem - Existence and Uniqueness for Linear Equations Suppose p , q , f are continuous on an open interval I and a I . Then, given any b , b 1 , y '' + p H x L y ' + q H x L y = f H x L has a unique solution on I that satisfies the initial conditions y H a L = b and y ' H a L...
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