Page 553 - Modelling in Transport Phenomena A Conceptual Approach

B.l. FIRST-ORDER EQUATIONS WITH EXACT SOLUTIONS 533

If M dx + N dy is an exact differential, then the differential equation
Mdx + Ndy = 0 (B. 1-6)

is called an exact differential equation. Since an exact differential can be expressed
in the form of a total differential d4, then
Mdx + Ndy = d4 = 0 (B.l-7)

and the solution can easily be obtained as

+=C (B. 1-8)

where C is a constant.

Example B.2 Solve the following differential equation

(42 - 3y)d~ + (1 - 32)dy = 0

Solution

Notethat M=4~-3yandN=1-3~. Since
OM aN
--
--- - -3
ay ax
the diffemntial equation is exact and can be expmssed in the form of a total diger-
ential d4,

From Eq. (2) we see that

Partial integration of Eq. (3) with respect to x gives

4 = 2x2 - 3~ + h(y) (5)

Substitution of Eq. (5) into Eq. (4) yields
_-
dh
-1
dY