Using differential equation of linear S. H.M, obtain the expression for (a) velocity in S. H.M., (b) acceleration in S. H.M.

Let , ,  and denote three constants (with the requirement that .) The following assumes that the mass of this object is . Assume that denotes the position of the object at time .

(a) .

(b)

Explanation:

The differential equation for a simple harmonic motion might take the following form:

.

The minus sign on the right-hand side highlights the fact that the displacement and acceleration of the object should be in opposite directions.

Notice how this equation is in the form of a homogeneous second-order ODE:

Let be a constant. One possible solution to this homogeneous second-order ODE would be in the form , such that whereas .

Substitute into the original ODE to obtain:

.

Rearrange the equation and solve for .

Notice that . Hence, it must be true that . Solve for given that :

, where is the imaginary unit.

The two particular solutions for the ODE would be:

and .

Apply Euler's Formula to rewrite both solutions in terms of trigonometric functions:

.

.

The general solution would be in the form:

,

Where and are constants (not necessarily real numbers.)

Since position is supposed to assume a real value for any real , set to a multiple of such that the general solution is real-valued:

.

Differentiate to obtain general expressions for velocity (first derivative) and acceleration (second derivative.)

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