Determining eigenfunctions + arbritary value constant

[JamesGoh]

When we determine an eigenfunction of a given differential equation, is it necessary to include the arbirtary value in front of the solution ?

If not, is it because of the term's arbritary nature which means we can choose to include/reject it from the determined eigenfunction ?

 

[WisheDeom]

If a differential equation allows for multiple solutions, the most general solution may contain an arbitrary term. This is always the case for linear equations, for example.

If, in addition, boundary conditions are given, then the arbitrary constant is set by those conditions.

As a basic example, the equation

$\frac{dx}{dt} + x = 0$

has a general solution

$x(t) = C e^{-t}$

but if an initial condition, e.g. $x(0) = 1$ is given, then the constant is set, and the particular solution for this inital value problem is

$x(t) = e^{-t}$