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AMath350.W13.A4.Sol

# B let y3 be the solution on 0 1 of the ivp x2 y 00 y 0

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Unformatted text preview: n (0, 1) of the IVP x2 y 00 + y 0 + xy = 0; y (1) = 2, y 0 (1) = 0. Find constants c1 and c2 such that y3 = c1 y1 + c2 y2 . Sol’n: We know that such constants exist, since y1 and y2 are a fundamental solution set. Diﬀerentiation tells us that 0 0 0 y3 = c1 y1 + c2 y2 AMATH 350 Assignment #4 Solutions- Winter 2013 Page 5 and setting x = 1 gives y3 (1) = c1 y1 (1) + c2 y2 (1) and 0 0 0 y3 (1) = c1 y1 (1) + c2 y2 (1). That is, 2 = c1 and 0 = c1 c2 , so c 1 = c 2 = 2. 7. (Considering the functions f (x) = x |x| and g (x) = x2 .) a) Solution: On (0, 1), f (x) = g (x), and so the functions are linearly dependent. On ( 1, 0), f (x) = g (x), and so the functions are linearly dependent on this interval as well. On the full interval ( 1, 1), consider the expression k1 f (x) + k2 g (x) = 0. For x > 0 this becomes k1 x2 + k2 x2 = 0 =) k1 = k2 , but for x < 0 we have instead k1 x2 + k2 x2 = 0 =) k1 = k2 , and so the only choice which works for all values of x 2 ( 1, 1) is k1 = k2 = 0 . Therefore f (x) and g (x)...
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