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The point x = 0 is a regular singular point of the given differential equation. 4x²y" - xy' + (x² + 1) y = 0 Substitute y = 2 n=0 x + into the differential equation and collect terms to rewrite as a single power series. r 4x²y² - xy² + (x²+1)y - (g(x-1) (4-1) )x+(₂(4²2²+3) ²¹ (G₂(k+r− 1) (4k+ Ár− 1) + C₂-2)*** = • 4r r+1 + X X 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.) k=2

The point x = 0 is a regular singular point of the given differential equation. 4x²y" - xy' + (x² + 1) y = 0 Substitute y = 2 n=0 x + into the differential equation and collect terms to rewrite as a single power series. r 4x²y² - xy² + (x²+1)y - (g(x-1) (4-1) )x+(₂(4²2²+3) ²¹ (G₂(k+r− 1) (4k+ Ár− 1) + C₂-2)*** = • 4r r+1 + X X 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.) k=2

Advanced Engineering Mathematics
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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The point x = 0 is a regular singular point of the given differential equation. 4x²y" - xy' + (x² + 1) y = 0 Substitute y = [ n=0 r= 1. cx 4 4x²y" - xy' + (x2² + 1) y = (Co(r-1) (4-1) )x² - (C₁ (4x² + 3r) )x¹+ (G₂ (k + r − 1) (4k + 4r − 1) + C₂_2 + 1 +r X X 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.) into the differential equation and collect terms to rewrite as a single power series. - Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on (0, ∞). Oy = C₁x¹/4(1- 1 1 x² 10 + x² + ...) + C₂x(1 - 112 x ² + 1 840 14 Oy = C₁x¹/4(1-x². 520 1 840 Oy=C₁x¹/4(1-22 Ⓒ y = C₁x¹/4(1- Oy = C₁x¹/4(1-1x² + 26 1 10 + x4 + 1 1672 x² + 1 520 1 1352 ...) + C₂x(1 - 1,2 26 12 ..) + C₂ x (1 - 11/10 * *² + ...) + C₂x(1 - 12/12 ² + x4 + 1 x4 + 1352 1 520 1 1672 1 840 ...) + C₂ x (1 - 1/24 x² + k=2 x + Ck-2x+r X
Transcribed Image Text:The point x = 0 is a regular singular point of the given differential equation. 4x²y" - xy' + (x² + 1) y = 0 Substitute y = [ n=0 r= 1. cx 4 4x²y" - xy' + (x2² + 1) y = (Co(r-1) (4-1) )x² - (C₁ (4x² + 3r) )x¹+ (G₂ (k + r − 1) (4k + 4r − 1) + C₂_2 + 1 +r X X 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.) into the differential equation and collect terms to rewrite as a single power series. - Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on (0, ∞). Oy = C₁x¹/4(1- 1 1 x² 10 + x² + ...) + C₂x(1 - 112 x ² + 1 840 14 Oy = C₁x¹/4(1-x². 520 1 840 Oy=C₁x¹/4(1-22 Ⓒ y = C₁x¹/4(1- Oy = C₁x¹/4(1-1x² + 26 1 10 + x4 + 1 1672 x² + 1 520 1 1352 ...) + C₂x(1 - 1,2 26 12 ..) + C₂ x (1 - 11/10 * *² + ...) + C₂x(1 - 12/12 ² + x4 + 1 x4 + 1352 1 520 1 1672 1 840 ...) + C₂ x (1 - 1/24 x² + k=2 x + Ck-2x+r X
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