EGR 509 ADVANCED DIFFERENTIAL EQUATIONS FOR ENGINEERS Summer 2002
Problem set #7
Radius of Convergence.
Determine the radius of convergence of the following series. (Show the details of your work.)
1. m



0
( m

1 ) mx m
2.

 m

0
1
3 m
( x

3 )
2 m
Text 3.8
: 3, 5
Text A4 : 1, 5, 7, 15 (pg. 538)
Note: The correct answer for problem A4-15 is y = 1

1
2 x
2 
1
6
Text A5 : 1, 3, 5 (pg. 551) x
3
+
1
40 x
5
+ ...
Problem set #8
Find solutions of the following differential equations by the Frobenius Method.
. Show the details of your work. Try to identify the series as expansions of known functions.
1.
x(1 – x)y” + 2(1 – 2x)y’ – 2y = 0
Ans: y
1
=
1 a
0
 x
, y
2
= a
0 x
=
B x
; Note: the constant a
0
in y
1 is not the same as a
0 in y
2
2.
xy” + 2y’ + xy = 0
Ans: y
1
= a
0 sin( x x )
, y
2
=
B cos( x ) x
3.
2x(x – 1)y” – (x + 1)y’ + y = 0
Ans: y
1
= a
0
(1 + x ), y
2
= Bx
1/2
Differential Equations Reducible to Bessel’s Equation.
Use the indicated substitutions and find a general solution in terms of J p
and J
-p
or indicate why these functions do not give a general solution. Show the details of your work.
4. x
2 y” + xy’ + (4x 4
-
1
4
)y = 0 (x
2
= z)
5. 4x
2 y” + 4xy’ + (x -
1
)y = 0 (x
1/2
= z)
36
6. 9x
2 y” + 9xy’ + (36x 4
- 16)y = 0 (x
2
= z)
Text 4.2
: 3, 5, 10, 11, 12

EGR 509 ADVANCED DIFFERENTIAL EQUATIONS FOR ENGINEERS Summer 2002
Problem set #9
Fourier-Legendre Series.
Develop in terms of Legendre polynominals:
3. x
5
,
3
Ans:
7
4
P
1
(x) +
9
8
P
3
(x) +
63
P
5
(x)
4.
15 – 42x 2 + 35x 4 ,
5.
y” + x 2 y = 0 (y = ux
1/2
,
1
2 x
2
= z),
Ans:
Some Further
Differential Equations Reducible to Bessel’s Equation.
Use the indicated substitutions and find a general solution in terms of Bessel functions. Show the details of your work.
Ans:
8(P
4
y = x
(x) - P
2
(x) +P
0
(x))
1/2
{c
1
J
1/4
(x
2
/2) + c
2
Y
1/4
(x
2
/2)}
6.
y” + k 2
xy = 0 (y = ux
1/2
,
2
3 kx
3/2
= z), Ans: y = x
1/2
{c
1
J
1/3
(2kx
3/2
/3) + c
2
Y
1/3
(2kx
3/2
/3)}
7.
y” + k 2
x
2 y = 0 (y = ux
1/2
,
1
2
1, 3, 5, 7, 13 (Text: Asmar) kx
2
= z), Ans: y = x
1/2
{c
1
J
1/4
(kx
2
/2) + c
2
Y
1/4
(kx
2
/2)}
4.4
5.2 1, 3, 5, 11 (Text: Asmar)