Linear beam theory allows several alternatives to solving the deflection differential equation from scratch.

Superposition

When the beam response is linear, the deflection caused by several loads is the sum of the deflections caused by each load separately.

A complicated loading case can therefore be decomposed into simpler cases with known solutions and recombined afterward.

Moment-area method

From

$$EI\frac{d^2y}{dx^2}=M(x),$$

the change in beam slope between two points equals the area under the $M/EI$ diagram between them.

The displacement of one tangent relative to another is related to the first moment of that area.

Choosing a method

Double integration is systematic and works directly from $M(x)$. Superposition is efficient when standard component solutions are available. Moment-area reasoning can make geometric relationships especially clear.

All of these methods describe the same linear elastic beam model; they differ only in how the governing relation is used.