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Unformatted text preview: Exercises 6.1
11. Let y1 = x−1 , y2 = x1/2 , and y3 = x. We want to ﬁnd constants c1 , c2 , and c3 such that c1 y1 + c2 y2 + c3 y3 = c1 x−1 + c2 x1/2 + c3 x = 0, for all x on the interval (0, ∞). This equation must hold if x = 1, 4, or 9 (or any other values for x in the interval (0, ∞)). By plugging these values for x into the equation above, we see that c1 , c2 , and c3 must satisfy the three equations c1 + c2 + c3 = 0 , c1 + 2 c2 + 4 c3 = 0 , 4 c1 + 3 c2 + 9 c3 = 0 . 9 Solving these three equations simultaneously yields c1 = c2 = c3 = 0. Thus, the only way for c1 x−1 + c2 x1/2 + c3 x = 0 for all x on the interval (0, ∞), is for c1 = c2 = c3 = 0. Therefore, these three functions are linearly independent on (0, ∞). 13. A linear combination, c1 x + c2 x2 + c3 x3 + c4 x4 , is a polynomial of degree at most four, and so, by the fundamental theorem of algebra, it cannot have more than four zeros unless it is the zero polynomial (that is, it has all zero coeﬃcients). Thus, if this linear combination vanishes on an interval, then c1 = c2 = c3 = c4 = 0. Therefore, the functions x, x2 , x3 , and x4 are linearly independent on any interval, in particular, on (−∞, ∞). 15. Since, by inspection, r = 3, r = −1, and r = −4 are the roots of the characteristic equation, r 3 + 2r 2 − 11r − 12 = 0, the functions e3x , e−x , and e−4x form a solution set. Next, we check that these functions are linearly independent by showing that their Wronskian is never zero. e3x W e ,e ,e
3x −x −4x e−x −e
−x e−4x −4e
−4x 1 =e e e
3x −x −4x 1 1 1 = −84e−2x , 16 (x) = 3e 3x 3 −1 −4 9 9e3x e−x 16e−4x which does not vanish. Therefore, {e3x , e−x , e−4x } is a fundamental solution set and, by Theorem 4, a general solution to the given diﬀerential equation is y = C1 e3x + C2 e−x + C3 e−4x . 343 ...
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.
 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations

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