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# ch03_7 - Ch 3.7 Variation of Parameters Recall the...

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Ch 3.7: Variation of Parameters Recall the nonhomogeneous equation where p , q, g are continuous functions on an open interval I . The associated homogeneous equation is In this section we will learn the variation of parameters method to solve the nonhomogeneous equation. As with the method of undetermined coefficients, this procedure relies on knowing solutions to homogeneous equation. Variation of parameters is a general method, and requires no detailed assumptions about solution form. However, certain integrals need to be evaluated, and this can present difficulties. ) ( ) ( ) ( t g y t q y t p y = + + 0 ) ( ) ( = + + y t q y t p y

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Variation of Parameters (1 of 6) We seek a particular solution to the equation below. We cannot use method of undetermined coefficients since g ( t ) is a quotient of sin t or cos t , instead of a sum or product. Recall that the solution to the homogeneous equation is To find a particular solution to the nonhomogeneous equation, we begin with the form Then or t y y csc 3 4 = + t c t c t y C 2 sin 2 cos ) ( 2 1 + = t t u t t u t y 2 sin ) ( 2 cos ) ( ) ( 2 1 + = t t u t t u t t u t t u t y 2 cos ) ( 2 2 sin ) ( 2 sin ) ( 2 2 cos ) ( ) ( 2
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ch03_7 - Ch 3.7 Variation of Parameters Recall the...

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