# m238sum08t2s - Mathematics 238 test two solutions Tuesday...

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Unformatted text preview: Mathematics 238 test two solutions Tuesday, July 22, 2008 1–7 . Find the characteristic equation & the general (real-valued) solution of each of the following linear homogeneous differential equations. 1 . y 00 + 25 y + 100 y = 0 r 2 + 25 r + 100 = 0 = ⇒ ( r + 20)( r + 5) = 0 = ⇒ r =- 20 ,r =- 5 y = c 1 e- 20 t + c 2 e- 5 t 2 . y 00 +20 y +100 y = 0 r 2 +20 r +100 = 0 = ⇒ ( r +10)( r +10) = 0 = ⇒ r =- 10 ,r =- 10 y = c 1 e- 10 t + c 2 te- 10 t 3 . y 00 + 12 y + 100 y = 0 r 2 + 12 r + 100 = 0 = ⇒ r =- 6 ± 8 i (from quadratic formula) y = e- 6 t ( c 1 cos 8 t + c 2 sin 8 t ) 4 . y 00 + 100 y = 0 r 2 + 100 = 0 = ⇒ r = ± 10 i y = c 1 cos 10 t + c 2 sin 10 t 5 . y (4) + 100 y (2) = 0 r 4 + 100 r 2 = 0 = ⇒ r 2 ( r 2 + 1) = 0 = ⇒ r = 0 ,r = 0 ,r = ± 10 i y = c + c 1 t + c 2 cos 10 t + c 3 sin 10 t 6 . ( D + 6)( D- 3) 2 ( D 2 + 5) y = 0 ( r + 6)( r- 3) 2 ( r 2 + 5) = 0 r =- 6 ,r = 3 ,r = 3 ,r = ± √ 5 y = c 1 e- 6 t + c 2 e- 3 t + c 3 te- 3 t + c 4 cos √ 5 t + c 5 sin √...
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m238sum08t2s - Mathematics 238 test two solutions Tuesday...

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