HW 2 solution - EGM 4313 S08 Homework Solution #2 EGM 4313...

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EGM 4313 S08 Homework Solution #2 1/7 EGM 4313 S08 - Homework#2 1. Problem Set 2.7, pp 83, #3: Find a (real) general solution. Which rule are you using? (Show each step of your calculation.) x x e e y y 60 2 . 19 16 ' ' 4 + = e q 1 Solution: Step 1: General solution of the homogeneous ODE 0 16 ' ' = y y e q 2 Let x h ce y λ = e q 3 x h e c y 2 ' ' = e q 4 Sub. eq 3 and 4 into eq 2 0 ) 16 ( 2 = x ce (characteristic equation) 4 , 4 = x x h e c e c y 4 2 4 1 + = e q 5 Step 2: Apply the Sum rule for the nonhomogeneous ODE 2 1 p p p y y y + = e q 6 (a) Using the Basic rule for 1 p y Let x p e c y 3 1 = e q 7 x p e c y 3 1 ' ' = e q 8 (b) Using the Modification rule for 2 p y Let x p xe c y 4 4 2 = e q 9 x x p xe c xe c y 4 4 4 4 2 16 8 ' ' + = e q 1 0 Step 3: 2 1 p p h y y y y + + = x x x x x p p h xe c e c e c e c e c y y y y 4 3 4 4 3 4 2 4 1 2 1 16 8 16 16 ' ' ' ' ' ' ' ' + + + + = + + = eq 11
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EGM 4313 S08 Homework Solution #2 2/7 Sub eq 10 and 11 into eq 1 x x x x x x x x x x x e e xe c e c e c e c xe c e c e c e c e c 60 2 . 19 ) ( 16 16 8 16 16 4 4 4 3 4 2 4 1 4 4 4 4 3 4 2 4 1 + = + + + + + + + eq 12 Simplify eq 12 x x x x e e e c e c 60 2 . 19 8 15 4 4 4 3 + = + 4 . 2 ; 4 4 3 = = c c The real general solution x x x x p p h xe e e c e c y y y y 4 4 2 4 1 2 1 4 . 2 4 + + = + + =
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EGM 4313 S08 Homework Solution #2 3/7 2. Problem Set 2.8, pp.90, #3 Find the steady-state oscillation of the mass-spring system modeled by the given ODE.
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HW 2 solution - EGM 4313 S08 Homework Solution #2 EGM 4313...

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