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**Unformatted text preview: **Exercises 6.2
given in Problem 34. Expanding the determinant over its last column yields x 1 sin x cos x y y y = y W [x, sin x, cos x] − y cos x − sin x x 1 sin x cos x sin x 1 cos x − sin x sin x 0 − sin x − cos x 0 − cos x cos x − sin x sin x y x +y sin x 0 − cos x cos x sin x − 0 − sin x − cos x − y 0 − sin x − cos x 0 − cos x 0 − cos x sin x cos x − cos x sin x − sin x − cos x − cos x sin x cos x − sin x − cos x − cos x sin x − sin x − cos x sin x = −xy − y x +y x −y = −xy + y − xy + y = 0. EXERCISES 6.2: Homogeneous Linear Equations with Constant Coeﬃcients, page 331 1. The auxiliary equation r 3 + 2r 2 − 8r = 0 ⇒ r r 2 + 2r − 8 = r (r − 2)(r + 4) = 0 has the roots r = 0, 2, and −4. Thus a general solutions to the diﬀerential equation has the form y = c1 + c2 e2x + c3 e−4x . 3. The auxiliary equation for this problem is 6r 3 + 7r 2 − r − 2 = 0. By inspection we see that r = −1 is a root to this equation and so we can factor it as follows 6r 3 + 7r 2 − r − 2 = (r + 1)(6r 2 + r − 2) = (r + 1)(3r + 2)(2r − 1) = 0. Thus, we see that the roots to the auxiliary equation are r = −1, −2/3, and 1/2. These roots are real and non-repeating. Therefore, a general solution to this problem is given by z (x) = c1 e−x + c2 e−2x/3 + c3 ex/2 . 351 ...

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