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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
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. 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 Need Help? Watch It Read It
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 Need Help? Watch It Read It
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