347_pdfsam_math 54 differential equation solutions odd

# 347_pdfsam_math 54 differential equation solutions odd -...

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Exercises 6.1 11. Let y 1 = x 1 , y 2 = x 1 / 2 , and y 3 = x . We want to find constants c 1 , c 2 , and c 3 such that c 1 y 1 + c 2 y 2 + c 3 y 3 = c 1 x 1 + c 2 x 1 / 2 + c 3 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 c 1 , c 2 , and c 3 must satisfy the three equations c 1 + c 2 + c 3 = 0 , c 1 4 + 2 c 2 + 4 c 3 = 0 , c 1 9 + 3 c 2 + 9 c 3 = 0 . Solving these three equations simultaneously yields c 1 = c 2 = c 3 = 0. Thus, the only way for c 1 x 1 + c 2 x 1 / 2 + c 3 x = 0 for all x on the interval (0 , ), is for c 1 = c 2 = c 3 = 0. Therefore, these three functions are linearly independent on (0 , ). 13. A linear combination, c 1 x + c 2 x 2 + c 3 x 3 + c 4 x 4 , 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 c 1 = c 2 = c 3 = c 4 = 0. Therefore, the functions x , x 2 , x 3 , and x 4 are linearly independent on any interval, in particular, on (
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