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11. Show that if n E Z the only solutions of the differential equation p² F" (r) + rF'(r) – n²F(r) = 0, which are twice differentiable when r > 0, are given by linear combinations of rn and r-n when n + 0, and 1 and log r when n = 0.

11. Show that if n E Z the only solutions of the differential equation p² F" (r) + rF'(r) – n²F(r) = 0, which are twice differentiable when r > 0, are given by linear combinations of rn and r-n when n + 0, and 1 and log r when n = 0.

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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11. Show that if n E Z the only solutions of the differential equation p? F" (r) + rF'(r) – n²F(r) = 0, which are twice differentiable when r > 0, are given by linear combinations of p and r-" when n + 0, and 1 and log r when n = 0. [Hint: If F solves the equation, write F(r) = g(r)r", find the equation satisfied by g, and conclude that rg'(r) + 2ng(r) = c where c is a constant.]
Transcribed Image Text:11. Show that if n E Z the only solutions of the differential equation p? F" (r) + rF'(r) – n²F(r) = 0, which are twice differentiable when r > 0, are given by linear combinations of p and r-" when n + 0, and 1 and log r when n = 0. [Hint: If F solves the equation, write F(r) = g(r)r", find the equation satisfied by g, and conclude that rg'(r) + 2ng(r) = c where c is a constant.]
u = f1 1 u = 0 Au = 0 u = 0 u = fo Figure 11. Dirichlet problem in a rectangle
Transcribed Image Text:u = f1 1 u = 0 Au = 0 u = 0 u = fo Figure 11. Dirichlet problem in a rectangle
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