390_pdfsam_math 54 differential equation solutions odd

390_pdfsam_math 54 differential equation solutions odd - x...

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Chapter 6 Using (22) on page 329 and (28) on page 330 of the text, we conclude that the set of functions (with x as the independent variable) 1 ,x ,x 2 ,x 3 ,e x ,x e x ,e x cos 3 x, xe x cos 3 x, sin 3 x, xe x sin 3 x forms an independent solution set. A general solution is given then by c 1 + c 2 x + c 3 x 2 + c 4 x 3 + c 5 e x + c 6 xe x + c 7 e x cos 3 x + c 8 xe x cos 3 x + c 9 sin 3 x + c 10 xe x sin 3 x = c 1 + c 2 x + c 3 x 2 + c 4 x 3 +( c 5 + c 6 x ) e x +( c 7 + c 8 x ) e x cos 3 x +( c 9 + c 10 x ) e x sin 3 x. 7. (a) D 3 , since the third derivative of a quadratic polynomial is identically zero. (b) The function e 3 x + x 1isthesumo f e 3 x and x 1. The function x 1 is annihilated by D 2 , the second derivative operator, and, according to (i) on page 334 of the text, ( D 3)
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Unformatted text preview: x . Therefore, the composite operator D 2 ( D 3) = ( D 3) D 2 annihilates both functions and, hence, there sum. (c) The function x sin 2 x is of the form given in (iv) on page 334 of the text with m = 2, = 0, and = 2. Thus, the operator ( D 0) 2 + 2 2 2 = ( D 2 + 4 ) 2 annihilates this function. (d) We again use (iv) on page 334 of the text, this time with m = 3, = 2, and = 3, to conclude that the given function is annihilated by [ D ( 2)] 2 + 3 2 3 = ( D + 2) 2 + 9 3 . 386...
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