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VariationOfParameters

VariationOfParameters - Variation of parameters What to use...

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Variation of parameters What to use when undetermined coefficients won’t do February 28, 2008 Variation of parameters
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How could we solve y + y = sec t ? This is a constant coefficient linear second order inhomogeneous differential equation,but, the forcing term is not suitable for undetermined coefficients. Most forcing terms are not suitable for use of undetermined coefficients. There is another method that will work for linear second order equations providing that we know two independent solutions of the homogeneous equation. Variation of Parameters. Variation of parameters
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Variation of parameters Here is the setup: 1) We have the linear second order differential equation a ( t ) y + b ( t ) y + c ( t ) y = f ( t ) 2) We have two independent solutions of the homogeneous equations y 1 and y 2 . That is a ( t ) y 1 + b ( t ) y 1 + c ( t ) y 1 = a ( t ) y 2 + b ( t ) y 2 + c ( t ) y 2 = 0 3) We look for our particular solution in the form y p ( t ) = v 1 ( t ) y 1 ( t ) + v 2 ( t ) y 2 ( t ) Variation of parameters
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When we introduce the two unknowns v 1 , and v 2 , it is reasonable
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