4th Edition
Appendix K
K.1 Solutions to Differential Equations
K.3.A First-order Ordinary Differential Equations
Also see http://www.ucl.ac.uk/Mathematics/geomath/level2/deqn/
de8.html
dy
 f t y  gt 
dt
Integrating factor = exp

(K-15)
fdt
Multiply through the integrating

e

Collect term
f y dt
  fdt 
ye
 e  fdt gt
d 
 dt 



ye 
and divide by e
fdt
fdt
dy
 e  y  e  gt 
dt
fdt
  gt e 
fdt
dt  K1
 fdt 

 fdt
fdt
 fdt
y  e   gt e  dt  K1e 
(K-16)

Example A–1 Integrating Factor for Series Reactions

dy
 k 2 y  k1e k1t
dt
Integration factor  exp

 k2 dt  e k t
2
k t
k k t
 dye 2
 e k 2 t k1e k1t  k1e  2 1 
dt

k1
k k t
k k t
k2t
e y  k1  e  2 1  dt 
e  2 1   K1
k 2  k1
y

k1
e k1t  K1e k 2 t
k 2  k1
t 0

4thEd/AppnKinsert.doc
y 0
1

y

k1
e k1t  e k 2 t
k 2  k1

K.3.B Coupled First-order Linear Ordinary Differential Equations
with Constant
Coefficients

Also see http://www.mathsci.appstate.edu/~sjg/class/2240/finalss04/
Alicia.html.
Consider the following coupled set of linear first order ODE with constant
coefficients.
dx
(1)
dt
 ax  by
dy
 cx  dy
dt

(2)

t 0
x  x0
t 0

y  y0
Or in Matrix Notation
dx 
dt  a
dy  
c
 
dt 
b x
d 
 
y

The solution to these coupled equations is



x  Ae  Be t
t

(3)
and
y  K1e t  K 2 e  t

where
TrA  TrA 2  4DetA 
, TrA  a  d, DetA  ad  bc
,    


2


From the initial conditions
t = 0 x = x0 and y = y0 we get

x0  A  B
and

y 0  K1  K 2
differentiating equations (3) and (4) and evaluating the derivative at t = 0
dx 
   A  B  ax 0  by 0
dt t0

4thEd/AppnKinsert.doc

dy 
  K1  K 2  cx 0  dy 0
dt t0
2
(4)
We have four equations and four unknowns (The arbitrary constants of
integration A, B, K1 and K2) so we can eliminate these arbitrary constants of
integration.
If y0 = 0 then the solution takes the form
a   e  t  a   e t 

x  x 0 







y

cx 0
 
e  t 

3
4thEd/AppnKinsert.doc
cx 0
 
e t