5Second-order differential equations
IA Differential Equations
5.3 Linear equidimensional equations
Equidimensional equations are often called homogeneous equations, but this is
confusing as it has the same name as those with no forcing term. So we prefer
this name instead.
Definition (Equidimensional equation). An equation is equidimensional if it
has the form
ax
2
y
00
+ bxy
0
+ cy = f(x),
where a, b, c are constants.
To understand the name “equidimensional”, suppose we are doing physics
and variables have dimensions. Say
y
has dimensions
L
and
x
has dimensions
T
.
Then
y
0
has dimensions
LT
−1
and
y
00
has dimensions
LT
−2
. So all terms
x
2
y
00
,
xy
0
and y have the same dimensions.
Solving by eigenfunctions
Note that
y
=
x
k
is an eigenfunction of
x
d
dx
. We can try an eigenfunction
y
=
x
k
.
We have
y
0
=
kx
k−1
and thus
xy
0
=
kx
k
=
ky
; and
y
00
=
k
(
k −
1)
x
k−2
and
x
2
y
00
= k(k − 1)x
k
.
Substituting in, we have
ak(k − 1) + bk + c = 0,
which we can solve, in general, to give two roots
k
1
and
k
2
, and
y
c
=
Ax
k
1
+
Bx
k
2
.
Solving by substitution
Alternatively, we can make a substitution z = ln x. Then we can show that
a
d
2
y
dz
2
+ (b − a)
dy
dz
+ cy = f(e
z
).
This turns an equidimensional equation into an equation with constant coeffi-
cients. The characteristic equation for solutions in the form
y
=
e
λz
is of form
a
2
λ
2
+ (
b −a
)
λ
+
c
= 0, which we can rearrange to become
aλ
(
λ −
1) +
bλ
+
c
= 0.
So λ = k
1
, k
2
.
Then the complementary function is y
c
= Ae
k
1
z
+ Be
k
2
z
= Ax
k
1
+ Bx
k
2
.
Degenerate solutions
If the roots of the characteristic equation are equal, then
y
c
=
{e
kz
, ze
kz
}
=
{x
k
, x
k
ln x}
. Similarly, if there is a resonant forcing proportional to
x
k
1
(or
x
k
2
),
then there is a particular integral of the form x
k
1
ln x.
These results can be easily obtained by considering the substitution method
of solving, and then applying our results from homogeneous linear equations
with constant coefficients.