HW6 - Physics2210.Homeworkassignment6. B 8.5.4. y + 2 y + 2...

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1 Physics 2210. Homework assignment 6. B 8.5.4. y + 2 y + 2 y = 0 The characteristic equation: D 2 + 2 D + 2 = 0 , with two roots 1 + i and 1 i . The solution is y = e x (Ae ix + Be -ix ) = e x (c 1 sinx + c 2 cosx) B 8.5.5. ( D 2 2 D + 1) y = 0 The characteristic equation: D 2 2 D + 1 = 0 , with a repeated root: 1 . The solution is y = e x (A + Bx) . B 8.5.10. y 2 y = 0 The characteristic equation: D 2 2 D = 0 , with two roots: 0 and 2 . The solution is y = Ae 2x + B . B 8.6.2. ( D 2) 2 y = 16 The complementary solution is y c = e 2x (A + Bx) . The particular solution is y p =4. The final solution is y = y c + y p = e 2x (A + Bx) +4 . B 8.6.4. (D + 1)(D 3)y = 24e 3x The complementary solution is y c = Ae -x + Be 3x . We assume that the particular solution is y p = Ce -3x , and substitute it to the ODE to determine C. (D + 1)(D 3)Ce 3x = 24e 3x . The left hand side is: C(D + 1)(-3e 3x 3e 3x ) = 6C(D + 1)e 3x = 6C(-3e 3x + e 3x ) =12Ce 3x . Therefore, C=2. The final solution is y = y c + y p = A e -x + Be 3x +2e -3x . B. 8.6.8.
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HW6 - Physics2210.Homeworkassignment6. B 8.5.4. y + 2 y + 2...

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