The point x = 0 is a regular singular point of the given differential equation. 3x²y" - xy + (x² + 1) y = 0 Substitute y = ] ) x + ² + ( Show that the indicial roots r of the singularity do not differ by an integer. (List the indicial roots below as a comma-separated list.) 3x²y" - xy' + (x² r= n=0 Oy= | = C₁x¹/³ (1 C Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on (0, ∞). ) y = C₁x¹/³(1 1 320 ..) + C₂x( 1 ⋅ ...) 1- Oy = C₁x¹/³ (1 +7 + into the differential equation and collect terms to rewrite as a single power series. + 1) y = Need Help? 1- 1- 1 8 x² 1 14 1 16 + +² 10 + Oy=C₁x¹1/³(1-1² + + Read It 1 392 + 4 1 Oy = C₁x¹/³(1-x² + 320x² + ...) + + 1 4 -X" + 896 440 + 1 4 0x² + ...) + Watch It ..) + C₂x(1 − . 1 16 x2 ..) + C₂x(1 - 1 1/2+² 10 2 1 896 + +4 + + + 1 440 1 320 + x² + ...) ₂x(1-1² + 140x² + ...) 10 1) xx+² ₂x(1-x² + 392x² + ...) 2 1 4 14
The point x = 0 is a regular singular point of the given differential equation. 3x²y" - xy + (x² + 1) y = 0 Substitute y = ] ) x + ² + ( Show that the indicial roots r of the singularity do not differ by an integer. (List the indicial roots below as a comma-separated list.) 3x²y" - xy' + (x² r= n=0 Oy= | = C₁x¹/³ (1 C Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on (0, ∞). ) y = C₁x¹/³(1 1 320 ..) + C₂x( 1 ⋅ ...) 1- Oy = C₁x¹/³ (1 +7 + into the differential equation and collect terms to rewrite as a single power series. + 1) y = Need Help? 1- 1- 1 8 x² 1 14 1 16 + +² 10 + Oy=C₁x¹1/³(1-1² + + Read It 1 392 + 4 1 Oy = C₁x¹/³(1-x² + 320x² + ...) + + 1 4 -X" + 896 440 + 1 4 0x² + ...) + Watch It ..) + C₂x(1 − . 1 16 x2 ..) + C₂x(1 - 1 1/2+² 10 2 1 896 + +4 + + + 1 440 1 320 + x² + ...) ₂x(1-1² + 140x² + ...) 10 1) xx+² ₂x(1-x² + 392x² + ...) 2 1 4 14
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:The point x = 0 is a regular singular point of the given differential equation.
3x2у" - ху' + (x? + 1)у %3D 0
+ r
Substitute y =
into the differential equation and collect terms to rewrite as a single power series.
n=0
3x?y" – xy' + (x² + 1) y =
|x' +
+ 1
+
xk + r
%3D
k=2
Show that the indicial roots r of the singularity do not differ by an integer. (List the indicial roots below as a comma-separated list.)
r =
Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on (0, 0).
1
1
Oy = C,x!/3(1
.4
+
X-
320
+ C,x( 1
x2
+
8
16
896
..) + c>x(1
.) + C>x(1 -
1*...) + C>x(1 - i0
y = C,x/3(1
,2
4
+
1
1
4
C2
+
X-
+
+
X-
-
14
392
10
440
1
1
1
Oy = C,x'
1/3
1
,2
4
+
C2X
4
+
16
896
8
320
Oy = C,x!/3(1
1 ,2
1
4
-x² +
,2
-x² +
8
+
-
320
440
1
+
440
* +...) + cx(1 -
1
+
392
2
Oy = Cqx/3(1
4
x2
14
4
+
%3D
10
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