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Problems (Particular and General Solution of Differential Equations)
In Problems 112, find a particular solution of the given differential equation. In Problems 1316, find the general solution of the given differential equation. In Problems 1720, find the particular solution of the initial value problem. 21 A massspring system mx" + bx' + kx = 689 cos (2f) has an external force of 689 cos (2t) dynes, spring constant k = 29, damping constant b =4, and mass m = 1 gm. Find the general solution for the motion of the spring and the steady state part of the solution. 22 A massspring system mx" + bx' + kx = 2 sin t has an external force of 2 sin t dynes, spring constant k = 24, damping constant b = 12, and mass m = 3gm. Find the general solution for the motion of the spring and the steady state part of the solution. 23 In the massspring system (5) my" + by' + ky = cos (ωt), where m, b, and k are positive, show that the steady state part of the solution has amplitude 24 In Problem 23, show that the frequency ω in the forcing term for which the steady state has the largest amplitude is and the largest amplitude is 1/b. This frequency ω is called the resonant frequency. 25 Using Problem 24, find the resonant frequency for the massspring system y" + 6y' + 25y = cos (ωt).


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