Oscillations problem with theta substitution

[prinnori]

Homework Statement

I attached a file containing the problem statement, because it is impossible to reproduce all the symbols.

Homework Equations

Also in the attached file.

The Attempt at a Solution

I tried to simply substitute theta but it turned out to be very messy...

 

[tiny-tim]

welcome to pf!

hi prinnori! welcome to pf!

I tried to simply substitute theta but it turned out to be very messy...

show us your full calculations, and then we'll see what went wrong, and we'll know how to help!

 

[prinnori]

θ(t)= A ( cos ωt + ε cos 3ωt );
d²θ/dt² = -ω² A (cos ωt + 9ε cos 3ωt);
θ³(t) = A³ ( cos³ωt + 3ε cos²ωt cos 3ωt + 3ε² cos ωt cos²3ωt + ε³ cos³ 3ωt ) =
= ... =
= A³/4 [ 3cos ωt (2ε²+ε+1) + cos 3ωt (3ε²+6ε+1) + 3cos 5ωt (ε²+ε) + 3ε² cos 7ωt + ε³ cos 9ωt]

Replacing all this in the equation won't help... I need a different approach but I can't think of anything else...

 

[tiny-tim]

hi prinnori!

i] you can ignore higher powers of A and ε

ii] you need to express cos3ωt in terms of cosωt and cos³ωt (and no sin)