Page 59 - Standard Handbook Of Petroleum & Natural Gas Engineering

48 Mathematics

the general solution is

y = u + c,u, + C,U, + . . . + C"U",
where u is any solution of the given equation and ul, uq, . . ., un form a
fundamental system of solutions to the homogeneous equation [E(x) t zero]. A
set of functions has linear independence if its Wronskian determinant, W(x),
# 0, where

UI up ... u,
u, up ... U"
W(x) = . . ... .
u;" u; ... u:

and m = n - lLh derivative. (In certain cases, a set of functions may be linearly
independent when W(x) = 0.)

The Laplace Transformation

The Laplace transformation is based upon the Laplace integral which transforms
a differential equation expressed in terms of time to an equation expressed in
terms of a complex variable B + jw. The new equation may be manipulated
algebraically to solve for the desired quantity as an explicit function of the
complex variable.
Essentially three reasons exist for the use of the Laplace transformation:
1. The ability to use algebraic manipulation to solve high-order differential
equations
2. Easy handling of boundary conditions
3. The method is suited to the complex-variable theory associated with the
Nyquist stability criterion [ 11.

In Laplace-transformation mathematics, the following symbols and variables
are used:
f(t) = a function of time
s = a complex variable of the form (O + jw)
F(s) = the Laplace transform of f, expressed in s, resulting from operating on
f(t) with the Laplace integral.
6: = the Laplace operational symbol, i.e., F(s) = S[f(t)].
The Laplace integral is defined as

6: = jo-e-"dt and so

6:[f(t)] = re-'"f(t)dt

Table 1-8 lists the transforms of some common time-variable expressions.