The point x = 0 is a regular singular point of the given differential equation. 3x²y" - xy + (x² + 1)y = 0 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, ∞). Oy = ₁x¹/3(1-x² 1,2 1 x4 320 1 x² 10 Oy=C₁x¹/³(1-1 14 Oy = C₁x¹/3(1-1x². Oy=C₁x¹/3(1- - + x² 1 10 + + x2 1 392 1 320 + + +4+ 440 v4 + 896 ...) + C₂x (1 -- 1 x² + x4 ...) + C₂ x (1 -. ..) + C₂x (1-. + 1 440 1 12 x4 10 440 1 1 y2 + x4 16 896 1 392 + + A + ..) + C₂x(1 − 1/1/1 x² + . 14 + + 1,2 + Oy=C₁x1/3(1-x²+*+...) ₂x(1-²20¹...) 16 x² + +
The point x = 0 is a regular singular point of the given differential equation. 3x²y" - xy + (x² + 1)y = 0 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, ∞). Oy = ₁x¹/3(1-x² 1,2 1 x4 320 1 x² 10 Oy=C₁x¹/³(1-1 14 Oy = C₁x¹/3(1-1x². Oy=C₁x¹/3(1- - + x² 1 10 + + x2 1 392 1 320 + + +4+ 440 v4 + 896 ...) + C₂x (1 -- 1 x² + x4 ...) + C₂ x (1 -. ..) + C₂x (1-. + 1 440 1 12 x4 10 440 1 1 y2 + x4 16 896 1 392 + + A + ..) + C₂x(1 − 1/1/1 x² + . 14 + + 1,2 + Oy=C₁x1/3(1-x²+*+...) ₂x(1-²20¹...) 16 x² + +
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.
·3x²y" - xy' + (x² + 1)y = 0
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.)
Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on (0, ∞).
1
1
Oy = C₁x¹/³(1 - 1x² + 320x4 + ...) +
₂x(1 - 10x² + ₁x² + ...)
8
440
○ y = C₁x¹/³(1 –
1 2
14
8
Oy=C₁x¹/3(1-1x² + 320
1
4
+
2
1/2 x ² + ...) + C₂x(1 - 11/10 x ² + 140x²
-X +
392
2
○y = C₁x¹/³(1-_¹_x² + _1_x4 +
-X
16
896
·) + ₂x(1 - 1²6x² +
1
1
1
Oy = C₁x¹/³(1-1x² + 4x4 + ...) + C₂x(1 - 4x² + 32 +4 + ...)
+4
10
14
...)
1
₁x² + ...)
896
1.2
1 4
...)
·.) + C₂x(1 - 1x² + 320x² + ...)
-X
8
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