4First-order differential equations
IA Differential Equations
4.1 The exponential function
Often, the solutions to differential equations involve the exponential function.
Consider a function f(x) = a
x
, where a > 0 is constant.
x
y
O
1
The derivative of this function is given by
df
dx
= lim
h→0
a
x+h
− a
x
h
= a
x
lim
h→0
a
h
− 1
h
= λa
x
= λf(x)
where
λ
=
lim
h→0
a
h
− 1
h
=
f
0
(0) = const. So the derivative of an exponential
function is a multiple of the original function. In particular,
Definition (Exponential function).
exp
(
x
) =
e
x
is the unique function
f
satisfying f
0
(x) = f(x) and f(0) = 1.
We write the inverse function as ln x or log x.
Then if y = a
x
= e
x ln a
, then y
0
= e
x ln a
ln a = a
x
ln a. So λ = ln a.
Using this property, we find that the value of e is given by
e = lim
k→∞
1 +
1
k
k
≈ 2.718281828 ··· .
The importance of the exponential function lies in it being an eigenfunction of
the differential operator.
Definition (Eigenfunction). An eigenfunction under the differential operator
is a function whose functional form is unchanged by the operator. Only its
magnitude is changed. i.e.
df
dx
= λf
Example. e
mx
is an eigenfunction since
d
dx
e
mx
= me
mx
.