Definitions

# ordinary differential equation

Equation containing derivatives of a function of a single variable. Its order is the order of the highest derivative it contains (e.g., a first-order differential equation involves only the first derivative of the function). Because the derivative is a rate of change, such an equation states how a function changes but does not specify the function itself. Given sufficient initial conditions, however, such as a specific function value, the function can be found by various methods, most based on integration.

In those hierarchically organised churches of Western Christianity which have an ecclesiastical law system, an ordinary is an officer of the church who by reason of office has ordinary power to execute the church's laws. The term comes from the Latin word ordinarius. In Eastern Christianity, a corresponding officer is called a hierarch, which comes from the Greek word ἱεράρχης meaning "priestly ruler".

## Ordinary power

In canon law, the power to govern the church is divided into the power to make laws (legislative), enforce the laws (executive), and to judge based on the law (judicial). A person exercises power to govern either because the person holds an office to which the law grants governing power or because someone with governing power has delegated it to the person. Ordinary power is the former, while the latter is delegated power. The office with ordinary power could possess the governing power itself (proper ordinary power) or instead it could have the ordinary power of agency, the inherent power to exercise someone else's power (vicarious ordinary power).

The law vesting ordinary power could either be ecclesiastical law, i.e. the positive enactments that the church has established for itself, or divine law, i.e. the laws which the church believes were given to it by God. As an example of divinely instituted ordinaries, Roman Catholics believe that when Jesus established the Church he in turn established the episcopate and the Primacy of Simon Peter and endowed the offices with power to rule the Church. Thus, in the Roman Catholic Church, the office of successor of Simon Peter and the office of diocesan bishop possess their ordinary power even in the absence of positive enactments from the Church.

Many officers possess ordinary power but, due to their lack of ordinary executive power, are not called ordinaries. The best example of this phenomenon is the office of judicial vicar, a.k.a. officialis. The judicial vicar only has authority through his office to exercise the diocesan bishop's power to judge cases. Though the vicar has vicarious ordinary judicial power, he is not an ordinary because he lacks ordinary executive power. A vicar general, however, has authority through his office to exercise the diocesan bishop's executive power. He is therefore an ordinary because of this vicarious ordinary executive power.

## Catholic usage

### Local ordinaries/hierarchs

Local ordinaries are ordinaries over particular churches. The following officers are local ordinaries:

### Other ordinaries/hierarchs

Other officers are also ordinaries (Latin Church) or hierarchs (Eastern Churches), but not local ordinaries (Latin Church) or local hierarchs (Eastern Churches):

The Pope is local ordinary of Rome. He is also the ordinary, but not the local ordinary, of the Latin rite church. He is also the Ordinary of the Universal Church.