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(2). This problem involves solving an integral equation for y(x) using Fourier Transforms. The integral is in the form of a convolution. In your solution, you may not use tables, although you may use the basic properties of Fourier Transforms that were derived in class. The general equation is - 2 y(x) + ſg(x − t)y(t)dt = g(x) -00 A specific function g(x) will be given in part (b). The functions y(x) and g(x) go to zero as x + and their Fourier Transforms exist. 00 00 (a). Defining Y(k) = ſy(x)exp(−ikx)dx and G(k) = fg(x)exp(−ikx)dx, compute the 00- 00- Fourier Transform of the integral equation and solve for Y(k) in terms of G(k). . (b). Compute the Fourier Transform G(k) for the specific function g(x) = exp(−|x|) and substitute this into your solution for Y(k) from part (a). (c). Use the results of part (a) and (b) and invert the transform for Y(k) to solve the integral equation for y(x) for -∞

(2). This problem involves solving an integral equation for y(x) using Fourier Transforms. The integral is in the form of a convolution. In your solution, you may not use tables, although you may use the basic properties of Fourier Transforms that were derived in class. The general equation is - 2 y(x) + ſg(x − t)y(t)dt = g(x) -00 A specific function g(x) will be given in part (b). The functions y(x) and g(x) go to zero as x + and their Fourier Transforms exist. 00 00 (a). Defining Y(k) = ſy(x)exp(−ikx)dx and G(k) = fg(x)exp(−ikx)dx, compute the 00- 00- Fourier Transform of the integral equation and solve for Y(k) in terms of G(k). . (b). Compute the Fourier Transform G(k) for the specific function g(x) = exp(−|x|) and substitute this into your solution for Y(k) from part (a). (c). Use the results of part (a) and (b) and invert the transform for Y(k) to solve the integral equation for y(x) for -∞

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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(2). This problem involves solving an integral equation for y(x) using Fourier Transforms. The integral is in the form of a convolution. In your solution, you may not use tables, although you may use the basic properties of Fourier Transforms that were derived in class. The general equation is - 2 y(x) + ſg(x − t)y(t)dt = g(x) -00 A specific function g(x) will be given in part (b). The functions y(x) and g(x) go to zero as x + and their Fourier Transforms exist. 00 00 (a). Defining Y(k) = ſy(x)exp(−ikx)dx and G(k) = fg(x)exp(−ikx)dx, compute the 00- 00- Fourier Transform of the integral equation and solve for Y(k) in terms of G(k). . (b). Compute the Fourier Transform G(k) for the specific function g(x) = exp(−|x|) and substitute this into your solution for Y(k) from part (a). (c). Use the results of part (a) and (b) and invert the transform for Y(k) to solve the integral equation for y(x) for -∞<x<∞. 00 - 2 y(x) + √__ exp(− | x − t |)y(t)dt = exp(−| x |) (This is just the equation you get when you put the g(x) from part (b) into the equation part (a).)
Transcribed Image Text:(2). This problem involves solving an integral equation for y(x) using Fourier Transforms. The integral is in the form of a convolution. In your solution, you may not use tables, although you may use the basic properties of Fourier Transforms that were derived in class. The general equation is - 2 y(x) + ſg(x − t)y(t)dt = g(x) -00 A specific function g(x) will be given in part (b). The functions y(x) and g(x) go to zero as x + and their Fourier Transforms exist. 00 00 (a). Defining Y(k) = ſy(x)exp(−ikx)dx and G(k) = fg(x)exp(−ikx)dx, compute the 00- 00- Fourier Transform of the integral equation and solve for Y(k) in terms of G(k). . (b). Compute the Fourier Transform G(k) for the specific function g(x) = exp(−|x|) and substitute this into your solution for Y(k) from part (a). (c). Use the results of part (a) and (b) and invert the transform for Y(k) to solve the integral equation for y(x) for -∞<x<∞. 00 - 2 y(x) + √__ exp(− | x − t |)y(t)dt = exp(−| x |) (This is just the equation you get when you put the g(x) from part (b) into the equation part (a).)
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