final - MAP2302/2331 Exam3 Dr Sin No Calculators Answer the...

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Unformatted text preview: MAP2302/2331 Exam3 Dr Sin No Calculators. Answer the questions in the spaces provided on the question sheets. Please write your answers in full detail. If you run out of room for an answer, continue on the back of the page. Name: 1. (8 points) Let y 1 and y 2 be solutions of the equation ( * ) ay 00 + by + cy = 0 , where a , b and c are fixed constants. Show that for any constants C 1 and C 2 , the function φ = C 1 y 1 + C 2 y 2 is a solution of the above equation. Solution: Since y 1 and y 2 are solutions of (*) we have ay 00 1 + by 1 + cy 1 = 0 and ay 00 2 + by 2 + cy 2 = 0 . Multiplying these equations by C 1 and C 2 respectively, and adding the resulting equations yields the equation ( ** ) ( C 1 y 00 1 + C 2 y 00 2 ) + b ( C 1 y 1 + C 2 Y 2 ) + c ( C 1 y 1 + C 2 y 2 ) = 0 . Observe that C 1 y 1 + C 2 y 2 = ( C 1 y 1 + C 2 y 2 ) = φ and C 1 y 00 1 + C 2 y 00 2 = ( C 1 y 1 + C 2 y 2 ) 00 = φ 00 . Substitute these into (**) to get aφ 00 + bφ + cφ = 0 . This last equation shows that φ is a solution of (*). MAP2302/2331 Exam3 Dr Sin 2. (8 points) Consider two mass spring systems with mass 1kg and spring with stiffness 2N/m. The first is undamped and the second has a damping constant of 2N-sec/m....
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final - MAP2302/2331 Exam3 Dr Sin No Calculators Answer the...

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