Define linear S.H.M. obtain a differential equation of linear S.H.M?

Linear simple harmonic motion (SHM) is a one-dimensional motion about a fixed equilibrium position in which the restoring force is directly proportional to displacement and opposite in direction:

F = −k x, k > 0

Derivation of the Differential Equation

1. From Newton’s second law: m (d²x/dt²) = F = −k x
2. Rearranging: (d²x/dt²) + (k/m) x = 0
3. Let ω² = k/m, then: d²x/dt² + ω² x = 0

Equation of Linear SHM: d²x/dt² + ω² x = 0

General Solution

The solution of the equation is:

x(t) = A cos(ωt) + B sin(ωt)

or equivalently,

x(t) = C cos(ωt + φ)

where A, B (or C, φ) are constants determined by initial conditions.