A note on Linear Superposition of functions

Linear algebraic functions can be expressed as a sum of two other functions, given certain constraints. These constraints are typically the boundaries of the original function, from which the two secondary functions can be derived.

Consider a function f(x), defined in the interval [a,b]. The linearity theory helps define two additional functions such that

Several functions can be derived for f1(x) and f2(x) that satisfy the above condition. Considering the simplest linear case, a value for the constant (delta) can be derived with a few generalized assumptions. Each of the functions f1(x) and f2(x) can be constructed such that it meets one of the boundary requirements individually. Namely,

Substituting either one of these conditions for f1(x) or f2(x) into the original equation for f(x), an expression for the constant can be obtained: 

The original function f(x) can then be constructed as a linear sum of f1(x) and f2(x). An example is shown below for one such construction.This concept of linear superpositions come in handy while solving boundary value ODE’s.