Verifying Solutions to Newton's Equation for a Harmonic Oscillator

[terp.asessed]

Homework Statement

I am trying to solve the textbook questions, but the steps are not shown--any suggestions would be appreciated!:

1) Verify that x(t) = Asin (wt) + B cos(wt), where w = (k/m)^1/2 is a solution to Newton's equation for a harmonic oscillator.

2) Verify that x(t) = Csin(wt + Φ) is a solution to Newton's equation for a harmonic oscillator.

Homework Equations

Given above...

The Attempt at a Solution

1) I only have a faint idea, but don't know where to progress...

2) I think I am going in a right direction, but don't know if it is of "ENOUGH" verification:

sin(wt +Φ) = sin(wt)cosΦ + cos(wt)sinΦ, which I put into the x(t) function:
x(t) = Csin(wt)cosΦ + Ccos(wt)sinΦ
= c₁sin(wt) + c₂cos(wt)

∴ c₁= CcosΦ
c₂= CsinΦ

...do I need further proof?

 

[BvU]

1) The idea is that you fill in x(t) in "Newton's equation for a harmonic oscillator" . The equation is probably linear in x, so you can do the terms one by one and you can forget the constants A and B

Same goes for 2). What you do in your attempt for 2) is convert a solution of type 2) into one of type 1. So once you've done 1) properly, you are also done with 2).

You don't say, but I suppose in your context, Newton's equation for a harmonic oscillator is something like $m\ddot x + k x = 0$ ?

 

[haruspex]

You don't say, but I suppose in your context, Newton's equation for a harmonic oscillator is something like $m\ddot x + k x = 0$ ?

... and that's the equation that should have been posted as "relevant equations".