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More review problems MATH 1220, Spring 2005 (1) (2) (3) (4) (5) (§18.1) (§18.1) (§18.2) (§18.2) (§18.2) Solve the differential equation: Solve the differential equation: Solve the differential equation: Solve the differential equation: Using the Method of Variation y ′′ + 2y ′ − 8y = 0. (D3 − 4D2 + 6D)y = 0. y ′′ + 3y ′ − 4y = 4x2 − 6x − 10. y ′′ − 4y = cos 2x − sin 2x. of Parameters solve the following differential equation: y ′′ − 3y ′ + 2y = 5x + 2. (6) (§18.3) A 12 pound weight stretches a spring 2 inches. The weight is raised 2 inches and given an initial velocity of 2 feet per second upward. Find the equation of the motion. (7) (§18.3) A spring with constant k of 10 pounds per foot is loaded with a 32 pound weight and brought to equilibrium. It is the stretched 2 inches and released. Find the equation of the motion if the damping force is proportional to twice the velocity. 1 2 Answers to more review problems MATH 1220, Spring 2005 (1) y = C1 e2x + C2 e−4x . √ √ (2) y = C1 + e2x (C2 sin 2x + C3 cos 2x). (3) y = C1 e−4x + C2 ex + 2 − x2 . 1 (4) y = C1 cos 2x + C2 sin 2x + x(cos 2x + sin 2x). 4 19 5 (5) y = C1 e2x + C2 ex + x + . 4 √2 √ √ 1 3 sin 8 3t. (6) y = − cos 8 3t − 6 12 1 1 −t cos 3t + sin 3t . (7) y = e 6 18