Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

Appendix C 267
Appendix C: Solution of inhomogeneous differential equations
C.1 General method
An inhomogeneous DE is a DE of the type
0 + () × = ()
(C.1)
In order to solve Eq. C.1, we first find a solution of the homogeneous DE
0 + () × =0
(C.2)
and then determine a particular solution of the inhomogeneous DE:
• Solution of the homogeneous DE by separation of variables:
0 + () × =0
0 = − () ×
(C.3)
(C.4)
y
ln
= − () (C.5)
= × − ()
(C.6)
• Determination of a particular solution of the inhomogeneoues DE by variation of
constant:
→ ()
(C.7)
0 () =
(C.8)
Insertion of () and 0 () into Eq. C.1 and integration gives
=
(C.9)
The general solution of Eq. C.1 is then given by
= +
(C.10)