Question

Q.3 (Applications of Linear Second Order ODE): Consider the 'equation of motion given by ODE d²x 102 +w²x = Focos(yt) where Fo and wty are constants. Without worrying about those constants, answer the questions (a)-(b). (a) Show that the general solution of the given ODE is [2 pts] Fo x(t) == xc + Ip = ci cos(wt) + C2 sin(wt) +- W2 - 92 cos(7t). (b) Find the values of ci and c2 if the initial conditions are x(0) = 0 and x'(0) = 0, so that the general solution in part (a) can be written explicitly as x(t) = -2 cos(7t) - [2 pts] For

Answer 1

(a) Given oDE u o Cos(YE) dt First we ind CF of ODE for ohich Now, aurilay cauatian is CoAnespeuding So G Coswt +c Sinwt CF Fo COsrt for PI 1 D+2 CoSrt Fo Fo Cos rt So, Complete oution (t) - CF + PI Sin wt+ fo Cosrt G Cos wt + a)

bTo find G&G we use Fo Coso (o)-0 0- C.I +GSino + Fo Now, (t)-Gwsinwt t wcoswt + (-Far) sinrt o(0)0+ wIto Substituching C4G in equati an chtain we x(t) Fo Coswt Fo Cosrt + wr ces yt - Fo Cosot 1

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