Γραμμική Διαφορική Εξίσωσις
- Είδος Διαφορικής Εξίσωσης
A differential equation is said to be linear if F can be written as a linear combination of the derivatives of y:
where ai(x) and r(x) continuous functions in x. Non-linear equations cannot be written in this form. The function r(x) is called the source term, leading to two further important classifications:
The equation that describes exponential decay is
or, by rearranging,
Integrating, we have
where C is the constant of integration, and hence
where the final substitution, N0 = eC, is obtained by evaluating the equation at t = 0, as N0 is defined as being the quantity at t = 0.
This is the form of the equation that is most commonly used to describe exponential decay. Any one of decay constant, mean lifetime, or half-life is sufficient to characterise the decay. The notation λ for the decay constant is a remnant of the usual notation for an eigenvalue. In this case, λ is the eigenvalue of the opposite of the differentiation operator with N(t) as the corresponding eigenfunction. The units of the decay constant are s−1.
Είναι μία Διαφορική Εξίσωση του τύπου
- Ly = f,
- ο Διαφορικός Τελεστής L is a Γραμμικός Τελεστής,
- y is the unknown function, and
- the right hand side f is a given function.
The linearity condition on L rules out operations such as taking the square of the derivative of y; but permits, for example, taking the second derivative of y. Therefore a fairly general form of such an equation would be
- where D is the differential operator d/dx (i.e. Dy = y' , D²y = y",... ), and the ai are given functions. Such an equation is said to have order n, the index of the highest derivative of f that is involved. (Assuming a possibly existing coefficient an of this derivative to be non zero, it is eliminated by dividing through it. In case it can become zero, different cases must be considered separately for the analysis of the equation.)
Αν και θα βρείτε εξακριβωμένες πληροφορίες
"Οι πληροφορίες αυτές μπορεί πρόσφατα
Πρέπει να λάβετε υπ' όψη ότι
- Μην κάνετε χρήση του περιεχομένου της παρούσας εγκυκλοπαίδειας
αν διαφωνείτε με όσα αναγράφονται σε αυτήν
- Όχι, στις διαφημίσεις που περιέχουν απαράδεκτο περιεχόμενο (άσεμνες εικόνες, ροζ αγγελίες κλπ.)