341h4 - M341 H4(S Zhang 1 EP 3.1 4 5 18 23 24 29 36...

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M341 H4 (S. Zhang) . 1. EP § 3.1: 4, 5, 18, 23, 24, 29, 36, 40 (Introduction, 2nd order linear equations) ans: 2. Verify that y 1 and y 2 are solutions. Then solve the IVP: y + 25 y = 0; y 1 = cos 5 x, y 2 = sin 5 x ; y (0) = 10 , y (0) = - 10 . ans: § 3.1-4. Plug in to verify solutions. y = C 1 y 1 + C 2 y 2 10 = C 1 cos 0 y = C 1 5( - sin 5 x ) + C 2 5 cos(5 x ) - 10 = C 2 5 cos 0 y = 10 cos 5 x - 2 sin 5 x 3. Verify that y 1 and y 2 are solutions. Then solve the IVP: y - 3 y + 2 y = 0; y 1 = e x , y 2 = e 2 x ; y (0) = 1 , y (0) = 0 . ans: § 3.1-5. Plug in to verify solutions. y = C 1 y 1 + C 2 y 2 1 = C 1 + C 2 y = C 1 e x + 2 C 2 e 2 x 0 = C 1 + 2 C 2 y = 2 e x - e 2 x 4. For nonlinear equations or non homgeneous equations, a linear combination of solutions may no longer a solution. Show that the given y is a solution, but cy is not a solution: y = x 3 ; yy = 6 x 4 . ans: § 3.1-18. yy = cx 3 · 6 cx = 6 c 2 x 4 = 6 x 4 So, when c = 1, y is a solution. When c 2 = 1, cy is not a solution. 5. ans: 6. Determine the linear dependence on the real line. f ( x ) = xe x ; g ( x ) = | x | e x ; ans: § 3.1-23. When x < 0, g ( x ) = - xe x = - f ( x ), but when x 0 g ( x ) = f ( x ). Because g ( x ) = Cf ( x ) for a fixed constant C , they are linearly independent. 7. Determine the linear dependence on the real line. f ( x ) = sin 2 x ; g ( x ) = 1 - cos 2 x ; ans: § 3.1-24. g ( x ) = - 2 sin 2 x = - 2 f ( x ) for all x , so they are linearly dependent. Using Wronskian: W ( f, g ) = f g f g = sin 2 x 1 - cos 2 x 2 sin x cos x sin 2 x = 0 8. Show that the given functions are solutions to the IVP.
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