The periodic outer force F' sin wt is applied to the mass-spring system from Task 2. The equation becomes y" = -3y – cy' + sin wt, c > 0. (a) Find a periodic solution of this equation in the form y(t) = a cos wt + b sin wt (the answer will depend on c, w). Prove that the amplitude of his solution is ((3 – w² )? + c²w? ) _0.5. (b) Fix c (you may assume that c is small) and find w = Wmaz > 0 such that the amplitude of this periodic solution reaches its maximum at w. Comment: you will see that for larger c, the amplitude monotonically decreases as wincreases, and thus there is no maximum point wmaz > 0. Hint: you may wish to denote v = w when searching for the maximum, but remember that v> 0. (C) For w 1, c = 2, write out the general solution of this equation (cv. Task 2a) and show that all solutions tend to the periodic solution as t → +0o.

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.