1 Linear ODEs: Questions
1. Find the general solution to the non-homogeneous equations
(a) y 00 5y 0 + 6y = 2ex
(b) x2 y 00 + 6xy 0 + 4y = 18x2 (This is a Euler equation)
2. Seek a series solution in powers of x for the following ODEs and find and the first three
non-zero terms of the solutions, along with the region of convergence of the series obtained:
(a) (1 x2 )y 00 2xy 0 + 34 y = 0 (Legendre’s equation of order 1
2
).
(b) (x2 1)y 00 6y = 0
(c) (1 x2 )y 00 2xy 0 + 12y = 0 (Legendre’s equation of order 3).
3. For the following ODEs Determine form of solution
f
egular
Y
singular (a) 4xy 00 + 2y 0 + y = 0, is it ordinary
singular
s (b) 2xy 00 + (3 x)y 0 y = 0,
roue find up to the first four non-zero terms for the two linearly independent solutions, as se-
ries about x = 0.
this
4. Given that
Z
x3 J0 (x)dx = x3 J1 (x) + 2x2 J0 (x) 4xJ1 (x),
deduce that if
1
X
2
x = ↵ m J0 ( m x)
m=0
for 0 < x < 1, where the m are given by
J0 ( m) = 0,
then
2 2
↵m = 3 J (
( m 4)
m 1 m)
Hint
The following properties of the Bessel functions will be useful
Z 1
xJ⌫ ( m x)J⌫ ( p x)dx = 0 for m 6= p,
0
Z 1
1 0 2
xJ⌫ ( m x)2 dx = (J⌫ ( m )) ,
0 2
d
J0 (x) = J1 (x)
dx
1
, 5g'tby 2e
a
y
tux eyn M 5Mt 6 0
CF P I
M 3and Mz 2 y
3 2x
Ae Be
Y
ex which is also in
integral already contains
since particular
Ce's this different
Gee X PI is
c F function try y y to
cextece Ce't Ce't Get AekBe't
y y Ice exe
y
Ce't Ge 6 Gex Lex
2CextCxe 5
he
Ice
exc
2cxex
Iii
zctzcy.ge g Ce
ex Ce
Act 2CK Z y y
seek Geek he
Ce
2ce Lex car
c ftp.I
General solution
yet Ae Be te
, the form
Euler
t tho t p so has solution
of
y
X
y
2
18
x'y Gag 44
y Ey't Ey
t 18
2
poop i pep 1 xp
Y
x
y g
sub into b
PCP 1 XPZ x a Gx pxp i that O
0
PCP 1 It 6pct 4xP
coefficients
Equating
PCP 1 top 4 0
p2 Pt Gpt 4 so
p2 y sp t 4 0
p l or p 4
since they are distinct Ax t Bit CF
y
particular integral try y Got Dat E
2C
y
Laced y
1. Find the general solution to the non-homogeneous equations
(a) y 00 5y 0 + 6y = 2ex
(b) x2 y 00 + 6xy 0 + 4y = 18x2 (This is a Euler equation)
2. Seek a series solution in powers of x for the following ODEs and find and the first three
non-zero terms of the solutions, along with the region of convergence of the series obtained:
(a) (1 x2 )y 00 2xy 0 + 34 y = 0 (Legendre’s equation of order 1
2
).
(b) (x2 1)y 00 6y = 0
(c) (1 x2 )y 00 2xy 0 + 12y = 0 (Legendre’s equation of order 3).
3. For the following ODEs Determine form of solution
f
egular
Y
singular (a) 4xy 00 + 2y 0 + y = 0, is it ordinary
singular
s (b) 2xy 00 + (3 x)y 0 y = 0,
roue find up to the first four non-zero terms for the two linearly independent solutions, as se-
ries about x = 0.
this
4. Given that
Z
x3 J0 (x)dx = x3 J1 (x) + 2x2 J0 (x) 4xJ1 (x),
deduce that if
1
X
2
x = ↵ m J0 ( m x)
m=0
for 0 < x < 1, where the m are given by
J0 ( m) = 0,
then
2 2
↵m = 3 J (
( m 4)
m 1 m)
Hint
The following properties of the Bessel functions will be useful
Z 1
xJ⌫ ( m x)J⌫ ( p x)dx = 0 for m 6= p,
0
Z 1
1 0 2
xJ⌫ ( m x)2 dx = (J⌫ ( m )) ,
0 2
d
J0 (x) = J1 (x)
dx
1
, 5g'tby 2e
a
y
tux eyn M 5Mt 6 0
CF P I
M 3and Mz 2 y
3 2x
Ae Be
Y
ex which is also in
integral already contains
since particular
Ce's this different
Gee X PI is
c F function try y y to
cextece Ce't Ce't Get AekBe't
y y Ice exe
y
Ce't Ge 6 Gex Lex
2CextCxe 5
he
Ice
exc
2cxex
Iii
zctzcy.ge g Ce
ex Ce
Act 2CK Z y y
seek Geek he
Ce
2ce Lex car
c ftp.I
General solution
yet Ae Be te
, the form
Euler
t tho t p so has solution
of
y
X
y
2
18
x'y Gag 44
y Ey't Ey
t 18
2
poop i pep 1 xp
Y
x
y g
sub into b
PCP 1 XPZ x a Gx pxp i that O
0
PCP 1 It 6pct 4xP
coefficients
Equating
PCP 1 top 4 0
p2 Pt Gpt 4 so
p2 y sp t 4 0
p l or p 4
since they are distinct Ax t Bit CF
y
particular integral try y Got Dat E
2C
y
Laced y
