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Q6B_230

Q6B_230 - y 00 4 y = cos x y(0 = 0 y π = 0 The general...

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Math 415 Quiz 6B A mass of 5 kg stretches a spring 10 cm. The mass is acted on by an external force of 10 sin( t/ 2) newtons and moves in a medium that imparts a viscous force of 2 newtons when the speed of the mass is 4 cm/sec. The mass is set in motion from its equilibrium position with an initial velocity of 3 cm/sec. Write down the initial value problem describing the motion of the mass, but do not solve it . The mass of five kilograms stretching the spring ten centimeters means that the spring constant is k = (5 * 9 . 8) / 0 . 1 = 490. The viscous medium makes the damping constant 2 / 0 . 04 = 50. So the differential equation is 5 u 00 + 50 u 0 + 490 u = 10 sin( t/ 2) , with initial conditions u (0) = 0 and u 0 (0) = 0 . 03. Solve the following boundary value problem or show that it has no solution.
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Unformatted text preview: y 00 + 4 y = cos x, y (0) = 0 , y ( π ) = 0 . The general solution to the homogeneous equation y 00 + 4 y = 0 is C 1 cos 2 x + C 2 sin 2 x . We can use the method of undetermined coeﬃcients to ﬁnd the speciﬁc solution for this non-homogeneous equation. Let Y ( x ) = A cos x + B sin x . Then Y 00 +4 Y = 3 A cos x +3 B sin x and hence A = 1 / 3 and B = 0. So our general solution here is C 1 cos 2 x + C 2 sin 2 x + 1 3 cos x . The ﬁrst boundary condition tells us that C 1 + 1 3 = 0, or C 1 =-1 3 . The second tells us that C 1-1 3 = 0, or C 1 = 1 3 . It is not possible to satisfy both conditions at the same time, so there is no solution to the equation. 1...
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