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The convolution product of two functions f and g is the function g * f defined by: g*f = g(t – v)f(v)dv This assignment presents another method of finding a particular solution to equations of the form ay'' + by' + cy = g(t) where a is not zero. For this, you will need two additional properties: 1. (y * g)'(t) = (y' * g)(t) + y(0)g(t) 2. (y * g)''(t) = (y" * 9)(t) + y'(0)g(t) + y(0)g'(t) A) Let y, (t) be the solution to ay'' + by' + cy = 0 that satisfies y, (0) = 0 and y, '(0) Show that y, * g is a particular solution to the non-homogeneous equation. B) Use the result above to find a particular solution to: i) y''+y = sec(t) ii) 2y'" + y' - y = e- sin(t)

The convolution product of two functions f and g is the function g * f defined by: g*f = g(t – v)f(v)dv This assignment presents another method of finding a particular solution to equations of the form ay'' + by' + cy = g(t) where a is not zero. For this, you will need two additional properties: 1. (y * g)'(t) = (y' * g)(t) + y(0)g(t) 2. (y * g)''(t) = (y" * 9)(t) + y'(0)g(t) + y(0)g'(t) A) Let y, (t) be the solution to ay'' + by' + cy = 0 that satisfies y, (0) = 0 and y, '(0) Show that y, * g is a particular solution to the non-homogeneous equation. B) Use the result above to find a particular solution to: i) y''+y = sec(t) ii) 2y'" + y' - y = e- sin(t)

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 convolution product of two functions f and g is the function g * f defined by: t g * f = g(t – v)f(v)dv This assignment presents another method of finding a particular solution to equations of the form ay'' + by' + cy = g(t) where a is not zero. For this, you will need two additional properties: 1. (y * g)'(t) = (y' * g)(t) + y(0)g(t) 2. (y * g)''(t) = (y" * g)(t) + y'(0)g(t) + y(0)g'(t) 1 A) Let ys (t) be the solution to ay'' + by' + cy 0 that satisfies Ys(0) = 0 and Ys'(0) Show that ys * g is a particular solution to the non-homogeneous equation. a B) Use the result above to find a particular solution to: i) y''+ y = sec(t) ii) 2y'’ + y' – y = e¯t sin(t)
Transcribed Image Text:The convolution product of two functions f and g is the function g * f defined by: t g * f = g(t – v)f(v)dv This assignment presents another method of finding a particular solution to equations of the form ay'' + by' + cy = g(t) where a is not zero. For this, you will need two additional properties: 1. (y * g)'(t) = (y' * g)(t) + y(0)g(t) 2. (y * g)''(t) = (y" * g)(t) + y'(0)g(t) + y(0)g'(t) 1 A) Let ys (t) be the solution to ay'' + by' + cy 0 that satisfies Ys(0) = 0 and Ys'(0) Show that ys * g is a particular solution to the non-homogeneous equation. a B) Use the result above to find a particular solution to: i) y''+ y = sec(t) ii) 2y'’ + y' – y = e¯t sin(t)
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