General solutions where f(x) = λ cosωx + µ sinωx (trig types)

General Solution of Harmonic Motions

Harmonic motion is a type of periodic oscillation, such as the swing of a pendulum or the vibration of a guitar string. In mathematics, this is typically modelled by trigonometric functions.
The equation f(x) = λ cosωx + µ sinωx is representative of harmonic motion, where λ and µ are the initial conditions of the motion (related to the initial position and velocity), ω is the frequency of the motion, and x is the independent variable, often time.

The general solution can be found by using the trigonometric identity for the sum of two angles: sin(a + b) = sin(a)cos(b) + cos(a)sin(b).
By comparing the structure of our equation with this identity, we can express our solution in the form R sin(ωx + α), where R is the amplitude of the motion and α is a phase shift.
To find the amplitude, R, apply the Pythagorean theorem: R = sqrt(λ² + µ²).
The phase shift, α, can be found by using the inverse tangent function: α = atan2(µ, λ), a special form of the arctan function that takes into consideration the correct quadrant of the angle.

The amplitude, R, represents the magnitude of the motion. In physical situations, this could represent the furthest distance a pendulum swings from its centre position.
The phase shift, α, gives us information on where in its cycle the motion starts. For instance, a springing motion may start from its extreme position (phase shift of 0) or from the centre position (phase shift of π/2).
The exact interpretation of these parameters depends on the context of the problem being solved.

The formulas and concepts used to solve the general harmonic motion equation are applicable in a wide array of disciplines such as physics, engineering, and music.
Understanding the solution helps to model, analyze, and predict behaviours of oscillatory systems like pendulums, oscillating springs, waves, and more.