Page 554 - Modelling in Transport Phenomena A Conceptual Approach

534 APPENDIX B. SOLUTIONS OF DWFERENTUL EQUATIONS

Integration of Eq. (6) gives the function h as

h=y+C (7)
where C is a constant. Substitution of Eq. (7) into Eq. (5) gives the function (b
as
(b = 2x2 - 3xy + y + c (8)

Hence, the solution is
2 x2 - 3xy + y = c* (9)
where Ca is a constant.

If the equation M dx + N dy is not exact, multiplication of it by some function
p, called an integrating factor, may make it an exact equation, i.e.,

(B.l-9)

For example, all thermodynamic functions except heat and work are state functions.
Although dQ is a path function, dQ/T is a state function. Therefore, l/T is an
integrating factor in this case.

B.1.3 Homogeneous Equations

A function f (2, y) is said to be homogeneous of degree n if

for all A. For an equation
Mdx+Ndy = 0 (B. 1-11)
if M and N are homogeneous of the same degree, the transformation

y=ux (B.1-12)

will make the equation separable.
For a homogeneous function of degree n, Euler's theorem states that

(B. 1-13)

Note that the extensive properties in thermodynamics can be regarded as homo-
geneous functions of order unity. Therefore, for every extensive property we can
write

(B.1-14)