Question

Can somebody explain me why the following equations are solutions for this differential equation? (Procedure)

Answer 1

Differential equation:\[\frac{ d ^{2}x }{ dt ^{2} }=-\omega ^{2}x\]
Solutions: \[x=Asen(\omega t)\]\[x=Acos (\omega t)\]\[x=Acos(\omega t + \theta _{o})\]Where A: amplitude, w: angular frequency, t: time and x: position.

Answer 2

Do you mean the procedure to solve it.

It can be solved using Laplace Transforms. However you can look at the equation and ask yourself what function of x can you take the second derivative and give you back the same function? Sine and Cosine do.

Answer 3

Yeah I just had a problem when doing the second derivative of those equations. Is it posible to solve the differential equation without suggesting the answer? I mean just plainly solve x in this equation\[\frac{d ^{2} x }{ dt ^{2} } =−ω^{2} x\]