Eigenfunction
An eigenfunction of an operator is unchanged up to multiplicative scaling by an eigenvalue. For , this means , so .
Definition 4.7 (Eigenfunction)
An eigenfunction of an operator is a function that is unchanged up to multiplicative scaling by the eigenvalue, under action of the operator.
In case of the differential operator, an eigenfunction satisfies
Radioactive decay
Example 4.15 (Radioactive Decay)
Consider the decay between three isotopes , with decay constants for and respectively.
Thus we have
and also
We shall try the particular integral of the form . Substituting it gives
Hence is the solution of the homogeneous equation . Thus
Thus, the general solution is
Now, if we were given the initial conditions , then
Hence,
Related
Stated in
- Definition 4.7 (Eigenfunction)§4.1 The Exponential Function
- Example 4.15 (Radioactive Decay)§4.3.2 Eigenfunction Forcing