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Q.3 (Applications of Linear Second Order ODE): Consider the ‘equation of motion’ given by ODE d2x...

Q.3 (Applications of Linear Second Order ODE): Consider the ‘equation of motion’ given by ODE d2x + ω2x = F0 cos(γt)

dt2
where F0 and ω ̸= γ 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] x(t) := xc + xp = c1 cos(ωt) + c2 sin(ωt) + (F0 / ω2 − γ2) cos(γt).

(b) Find the values of c1 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) =( F0/ ω2 − γ2) (cos(γt) − cos(ωt) ).

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