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) If F(ƒ) = ƒ(§) is the Fourier transform of f(x), show that the Fourier transform of x f(x) is F(xf) = ¸df dε' where is the Fourier transform variable. O Airy's equation is d²u xu = 0. dx² Use Fourier transforms and the result of part (a) to show that, by a suitable choice of a constant of integration, Airy's equation has the Airy function solution 1 Ai(x)= ½ cos (}§³+x) d£. π = COS

) If F(ƒ) = ƒ(§) is the Fourier transform of f(x), show that the Fourier transform of x f(x) is F(xf) = ¸df dε' where is the Fourier transform variable. O Airy's equation is d²u xu = 0. dx² Use Fourier transforms and the result of part (a) to show that, by a suitable choice of a constant of integration, Airy's equation has the Airy function solution 1 Ai(x)= ½ cos (}§³+x) d£. π = COS

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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3. (a) If F(ƒ) = ƒ (§) is the Fourier transform of f(x), show that the Fourier transform of x f(x) is F(xf) =-i ¸df 'dε' § where is the Fourier transform variable. (b) Airy's equation is d²u - xu = 0. dx² Use Fourier transforms and the result of part (a) to show that, by a suitable choice of a constant of integration, Airy's equation has the Airy function solution Ai(x) = = COS πT 1/14 √² cos ( ± €³ + x£) dε.
Transcribed Image Text:3. (a) If F(ƒ) = ƒ (§) is the Fourier transform of f(x), show that the Fourier transform of x f(x) is F(xf) =-i ¸df 'dε' § where is the Fourier transform variable. (b) Airy's equation is d²u - xu = 0. dx² Use Fourier transforms and the result of part (a) to show that, by a suitable choice of a constant of integration, Airy's equation has the Airy function solution Ai(x) = = COS πT 1/14 √² cos ( ± €³ + x£) dε.
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