Question

Question 4 (Unit 10) - 10 marks Consider the Airy equation d^2y/dx^2 - xy = 0. (a) Take the Fourier transform of this equation and show that the Fourier transform ?(k) satisfies d?/dk = ik^2?. (b) Verify that the general solution to the equation obtained in part (a) is ?(k) = Ce^(ik^3/3), where C is an arbitrary constant. (c) Hence show that a solution of the Airy equation is the Airy function Ai(x) = 1/? ?_0^? cos(kx + k^3/3) dk.

          Question 4 (Unit 10) - 10 marks

Consider the Airy equation

d^2y/dx^2 - xy = 0.

(a) Take the Fourier transform of this equation and show that the Fourier transform ?(k) satisfies

d?/dk = ik^2?.

(b) Verify that the general solution to the equation obtained in part (a) is

?(k) = Ce^(ik^3/3),

where C is an arbitrary constant.

(c) Hence show that a solution of the Airy equation is the Airy function

Ai(x) = 1/? ?_0^? cos(kx + k^3/3) dk.

Question 4 (Unit 10) - 10 marks

Consider the Airy equation

d^2y/dx^2 - xy = 0.

(a) Take the Fourier transform of this equation and show that the Fourier transform ?(k) satisfies

d?/dk = ik^2?.

(b) Verify that the general solution to the equation obtained in part (a) is

?(k) = Ce^(ik^3/3),

where C is an arbitrary constant.

(c) Hence show that a solution of the Airy equation is the Airy function

Ai(x) = 1/? ?0^? cos(kx + k^3/3) dk.

Added by Raymond D.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Supreeta N

09/04/2023

Step 1

First, we need to take the Fourier transform of the Airy equation. The Airy equation is given by: $$\frac{d^2y}{dz^2} = zy$$ Taking the Fourier transform with respect to $z$, we get: $$\mathcal{F}\left\{\frac{d^2y}{dz^2}\right\} = \mathcal{F}\{zy\}$$ Using the Show more…

Show all steps

Thanks for your feedback!

Consider the Airy equation d^2y/dx^2 - xy = 0. (a) Take the Fourier transform of this equation and show that the Fourier transform ỹ(k) satisfies dỹ/dk = ik^2ỹ. (b) Verify that the general solution to the equation obtained in part (a) is ỹ(k) = Ce^(ik^3/3), where C is an arbitrary constant. (c) Hence show that a solution of the Airy equation is the Airy function Ai(x) = 1/π ∫_0^∞ cos(kx + k^3/3) dk.