Page 555 - Modelling in Transport Phenomena A Conceptual Approach

B.1. FIRST-ORDER EQUATIONS WlTH EXACT SOLUTIONS 535

On the other hand, the intensive properties are homogeneous functions of order
zero and can be expressed as

(B. 1- 15)

Example B.3 Solve the following differential equation

xydx-(x2+y2)dy=0
Solution

Since both of the functions

M = xy
N = - (x2 + y2)
are homogeneous of degree 2, the tmnsformation

y=ux and dy=udx+xdu (3)
reduces the equation to the form

-+- 1+u2 du = 0 (4)
dx
X u3
Integration of Eq. (4) gives

xu=cexp(&) (5)

where C is an integration constant. Substitution of u = y/x into Eq. (5) gives the
solution as
1 x
Y = Cexp [- 2Y
(-)2]

B.1.4 Linear Equations

In order to solve an equation of the form
dY
- + P(.) Y = &(x) (B. 1-16)
dx
the first step is to find out an integrating factor, p, which is defined by

(B. 1- 17)