# Midterm2PracticeAnswers - k is even(b The equilibria are y...

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Answers for Midterm 2 Practice 1. (a) x sin( x ) + cos( x ) + C (b) - x 2 cos( x ) + 2 x sin( x ) + 2 cos( x ) + C (c) x ln( x ) - x + C (d) 16 3 ln(2) - 14 9 (e) 1 2 x (sin(ln( x )) - cos(ln( x ))) 2. (a) 1 5 ln ± ± ± x - 1 x +4 ± ± ± + C (b) 1 6 ln ± ± ± x - 3 x +3 ± ± ± + C (c) - 16 ( x - 3) + 5 ln | x - 3 | + C (d) 1 2 x 2 - ln( x 2 + 2)) + 2 2 arctan ² x 2 ³ + C (e) 1 2 ln ( 2 3 ) 3. (a) Convergent, = 1 2 (b) Convergent, = 0 (c) Divergent (d) Convergent, = - 2 (e) Convergent, = 2 4. (a) y ( x ) = arcsin ( x 2 ) + 3 (b) x ( t ) = Ce t 2 - 3 t + 1 (note this includes the constant solution x ( t ) = 1) (c) y ( x ) = 1 3 2+ e 2 3( x - 1) 2 - e 2 3( x - 1) (d) y ( x ) = e - e - x +1 (e) y ( x ) = - x 2 + 3 5. (a) The equilibria are y = for any integer k . The equilibria is locally stable when k is odd and unstable when
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Unformatted text preview: k is even. (b) The equilibria are y = 2, which is locally stable, and y = 3, which is unstable. (c) The equilibria are y = 1 2 , which is locally stable, and y = e 3 , which is unstable. 6. (a) u + v = 4-2-2 , u-v = -2-4 6 and u ˙ v =-8 (b) arccos ²-4 √ 91 ³ (c) 1 || u || u = 1 √ 14 1-3 2 , 1 || v || v = 1 √ 26 3 1-4 . (d) 3 x + y-4 z =-8 1...
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