2 (6) w7.2 Consider problem 3.5.52 from the point of view of undetermined coefficients, but let's take advantage of your

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2 6 W7 2 Consider Problem 3 5 52 From The Point Of View Of Undetermined Coefficients But Let S Take Advantage Of Your 1 (82.02 KiB) Viewed 35 times

2 (6) w7.2 Consider problem 3.5.52 from the point of view of undetermined coefficients, but let's take advantage of your knowledge of operator factorization and Euler's formula. We wish to solve L(y) = y" +9y = sin(3x) Since the roots of the characteristic polynomial p(r) = ² +9 are r = ±3i the term on the right, sin(3r) solves the homogeneous DE, and Case II of undetermined coefficients prescribes an undetermined coefficients test solution of the form Yp(x)= x(A cos(3x) + B sin(3x)) Carrying this procedure out works to find a particular solution, but is somewhat messy. Rather, factor L as L = D² +91 = [D+3iI] o [D-3iI] Recall from the previous homework that [DrI]f(x)er* = f'(x)ex 1 PROBLEM SET 6: DUE THURSDAY JUNE 30 11:59PM ON GRADESCOPE (a) Use factorization above, and the fact above to compute that L(re³ix) = 6ie³ix (b) Use Euler's formula and linearity to deduce from the real and imaginary parts of the formula in a that. L(x cos(3x)) = -6 sin(3x) L(x sin(3x)) = 6 cos(3x) (c) Use linearity and the result of (b) to deduce that a particular solution to the original differential equation is Yp(x) = x 6 x cos(3x)