This appendix illustrates the mathematical methods needed to solve equation (4.6), a first-order differential equation. Readers are referred to textbooks such as Chiang (1984) for more rigorous treatments.
We first re-arrange (4.6) into:(A.1)
To derive the general solution of (A.1), we first multiply it by an integrating factor, I, to make it an exact equation.(A.2)
Consider the following first-order differential equation,(A.3)which is said to be exact if and only if:(A.4)
Thus, for (A.2) to be exact,(A.5)
On solving, the integrating factor equals: