Vibrations and Waves · Problem 1-10

The superposition of periodic motions — A.P. French

Physics — Waves · differential-equations-shm

Problem

Verify that the differential equation

d^2y/dx^2 = -ky

has as its solution

y = A\cos(kx) + B\sin(kx)

where

A

and

B

are arbitrary constants. Show also that this solution can be written in the form

y = C\cos(kx + \alpha) = C\,\text{Re}[e^{j(kx+\alpha)}] = \text{Re}[(Ce^{j\alpha})e^{jkx}]

and express

C

and

\alpha

as functions of

A

and

B