Answer to Question #259237 in Differential Equations for Emmanuel

Question

If F = a cos wti + b sin wtj where a,b,c are constant, find

F*df/dt and prove that

d^2f/dt^2 + w^2f = 0

Please note, w refers to omega. Thank you very much.

Answer 1

f=a coswt i + b sin wt j

f×$\frac{df}{dt}$ =(a cos(wt) i + b sin(wt) j)×(-aw sin(wt) i + bw cos(wt) j)

=-a²w cos(wt) sin(wt)+b²wsin(wt)

Now,

$\frac{d^2f}{dt^2}+w^2f$

=-aw² cos(wt)i - bw² sin(wt)j+w² (a cos(wt) i + b sin(wt) j)

=-aw² cos(wt)i - bw² sin(wt)j+w²a cos(wt) i + bw² sin(wt) j

=0

Hence, proved.