If a mass is placed at the end of a spring and pulled down and then released, the mass-spring system oscillates with simple harmonic motion. The displacement x of the mass from its resting position is a function of time t, such that x(t) = c₁cos ωt + c₂sin ωt where c₁ and c₂ are arbitrary constants, and ω is a fixed constant.

Let W = {x(t): x(t) = c₁cos ωt + c₂sin ωt, c₁ & c₂ are arbitrary constants, and ω is a fixed constant}

Question

If a mass is placed at the end of a spring and pulled down and then released, the mass-spring system oscillates with simple harmonic motion. The displacement x of the mass from its resting position is a function of time t, such that x(t) = c₁cos ωt + c₂sin ωt where c₁ and c₂ are arbitrary constants, and ω is a fixed constant.

Let W = {x(t): x(t) = c₁cos ωt + c₂sin ωt, c₁ & c₂ are arbitrary constants, and ω is a fixed constant}

anonymous · Answer

a) Prove that the set W, with the standard definitions of vector addition and scalar multiplication, is a vector subspace.
b) Find a basis for the subspace W.
c) Use the properties of the basis vectors to prove that x = c₁cos ωt + c₂sin ωt is the general solution of the differential equation x'' + ω²x = 0

anonymous · Answer

(a) I'm good with this one
(b) I assume would just be done by inspection and the correct answer would be {cos ωt, sin ωt)
(c) I'm not sure how to use "the properties of basis vectors" to prove this