
Appendix A
Linear ordinary differential
equations
A.1 Linear differential equations
A univariate ordinary differential equation is one in which we, for given
functions a(t),f(t), try to find a function x(t) such that
x
(t)+a(t)x(t)=f(t), (A.1)
and for which we assume a start condition x(0) = x
0
. To solve such an
equation, we define a function A(t) such that A
(t)=a(t)andA(0) = 0, and
multiply equation A.1 by e
A(t)
. Using the product formula for differentiation
we can then rewrite the equation as
(e
A(t)
x(t))
= e
A(t)
f(t).
Integrating this relation from 0 to t shows that
e
A(t)
x(t)=x(0) +
t
0
e
A(s)
f(s) ds
from which we deduce that
x(t)=U(t, 0)x
0
+
t
0
U(t, s)f (s) ds,