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Unformatted text preview: Notes for Day : . : Variation of Parameters e Method of Variation of Parameters allows us to solve non-homogeneous second-order linear di erential equations of the form y + p ( t ) y + q ( t ) y = g ( t ) when g ( t ) is not in the right form to allow us to solve using the Method of Undetermined Coe cients. Both methods accomplish the same goal. Variation of Parameters is more versati e, but genera y onger, than Undetermined Coe cients. e basic idea Suppose we have already solved the homogeneous di erential equation: y + p ( t ) y + q ( t ) y = . Remember that we call this solution the complementary solution and use y C to represent it. e complementary solution takes the form: y C ( t ) = c y ( t ) + c y ( t ) , where c and c are constants. If instead, we allow functions u ( t ) and u ( t ) to be used in place of the constants, we get a particular solution instead: y P ( t ) = u ( t ) y ( t ) + u ( t ) y ( t ) , as long as the resulting function y P ( t ) solves the di erential equation.solves the di erential equation....
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- Spring '06