Chap
.
2
Second

Order Linear ODEs
Sec
.
2
.
1
Homogeneous Linear ODEs of Second Order
On pp. 4546 we extend concepts defined in Chap. 1, notably solution and homogeneous and
nonhomogeneous, to secondorder ODEs; take a look into Secs. 1.1 and 1.5 before you continue.
You will see in this section that a homogeneous linear ODE is of the form
y
u2032u2032
u002b
p
ue0a2
x
ue0a3
y
u2032
u002b
q
ue0a2
x
ue0a3
y
u003d
0.
An initial value problem for it will consist of
two
conditions, prescribing an
initial value
and an
initial slope
of
the solution, both at the same point
x
0
. But on the other hand, a general solution will now involve
two
arbitrary
constants for which some values can be determined from the two initial conditions. Indeed, a general solution
is of the form
y
u003d
c
1
y
1
u002b
c
2
y
2
where
y
1
and
y
2
are such that they cannot be pooled together with just one arbitrary constant remaining;
expressed technically,
y
1
and
y
2
are “linearly independent”, meaning that they are not proportional on the
interval on which a solution of the initial value problem is sought.
Problem Set 2
.
1
.
Page 52
5
.
General solution
.
Initial value problem
.
Substitution shows that
x
2
and
x
u2212
2
ue0a2
x
u2260
0
ue0a3
are solutions.
(The simple algebraic derivation of such solutions will be shown in Sec. 2.5 on p. 70.) They are linearly
independent (not proportional) on any interval not containing 0 (where
x
u2212
2
is not defined). Hence
y
u003d
c
1
x
2
u002b
c
2
x
u2212
2
is a general solution of the given ODE. Set
x
u003d
1 and use the initial conditions in
y
and
y
u2032
u003d
2
c
1
x
u2212
2
c
2
x
u2212
3
. This gives
y
ue0a2
1
ue0a3
u003d
c
1
u002b
c
2
u003d
11,
y
u2032
ue0a2
1
ue0a3
u003d
2
ue0a2
c
1
u2212
c
2
ue0a3
u003d
u2212
6.
The solution is obtained by inspection or elimination,
c
1
u003d
4,
c
2
u003d
7.
9
.
Linear independence
.
e
ax
/
e
u2212
ax
u003d
e
2
ax
is not constant, unless
a
u003d
0. Hence
y
u003d
c
1
e
ax
u002b
c
2
e
u2212
ax
with
a
u2260
0 can be a general solution of an ODE. You may verify that the ODE is
y
u2032u2032
u2212
a
2
y
u003d
0. A derivation
will be given in the next section.
11
.
Linear dependence
.
This follows by noting that ln
x
2
u003d
2 ln
x
. The problem is typical of cases in
which some functional relation is used to show linear dependence. Problem 13 is of the same kind.
21
.
Reduction to first order
.
The most general secondorder ODE is of the form
F
ue0a2
x
,
y
,
y
u2032
,
y
u2032u2032
ue0a3
u003d
0 . It
can be reduced to first order if [Case (A)]
x
does not occur explicitly, or if [Case B]
y
does not occur
explicitly. The ODE
y
u2032u2032
u002b
y
u2032
3
sin
y
u003d
0 is Case (A). To reduce it, set
z
u003d
y
u2032
u003d
dy
/
dx
and use
y
as the
independent variable. This can be done by using the chain rule, namely,
y
u2032u2032
u003d
dy
u2032
dx
u003d
dy
u2032
dy
u2219
dy
dx
u003d
dz
dy
z
u003d
u2212
z
3
sin
y
where the last equality follows by using the given ODE. Now divide by
u2212
z
3
and separate variables to get
u2212
dz
z
2
u003d
sin
y dy
.
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 Spring '12
 M.lee
 Sin, Cos, general solution, real solutions

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