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Consider a second-order linear non-homogeneous differential equation of the form: y" + p(x)y' + q(x)y = f(x) (1) where p(x), q(x) and f(x) are continuous functions on an interval I. The particular solution of this equations is given as: Yp(x) = u₁(x)y₁(x) + U₂(x)y2(x) (2) where y₁(x) and y2(x) are two linearly independent solutions of the corresponding homogeneous equation, and u₁ (x) and u2(x) are functions to be determined. (a) Show that the particular solution yp(x) can be expressed in terms of an integral involving the Green's function G(x, t): where G(x, t) is defined as: yp(x) = f*G(x,t)f(t) dt G(x, t) = Y1(t) y2(x) - Y₁(x) y2(t) W(t) where W(t) is the Wronskian of y₁ (t) and y2(t). (b) Verify that the integral form of yp(x) in Eq. (3) satisfies the following initial conditions: y(xo) 0, y'(xo) = 0 (c) In the light of previous parts, use the Green's function (Eq.3) to solve the following initial value problem: y" y = e²x with y(0) = 0 and y'(0) = 0 (3) (4)

Consider a second-order linear non-homogeneous differential equation of the form: y" + p(x)y' + q(x)y = f(x) (1) where p(x), q(x) and f(x) are continuous functions on an interval I. The particular solution of this equations is given as: Yp(x) = u₁(x)y₁(x) + U₂(x)y2(x) (2) where y₁(x) and y2(x) are two linearly independent solutions of the corresponding homogeneous equation, and u₁ (x) and u2(x) are functions to be determined. (a) Show that the particular solution yp(x) can be expressed in terms of an integral involving the Green's function G(x, t): where G(x, t) is defined as: yp(x) = f*G(x,t)f(t) dt G(x, t) = Y1(t) y2(x) - Y₁(x) y2(t) W(t) where W(t) is the Wronskian of y₁ (t) and y2(t). (b) Verify that the integral form of yp(x) in Eq. (3) satisfies the following initial conditions: y(xo) 0, y'(xo) = 0 (c) In the light of previous parts, use the Green's function (Eq.3) to solve the following initial value problem: y" y = e²x with y(0) = 0 and y'(0) = 0 (3) (4)

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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Consider a second-order linear non-homogeneous differential equation of the form: y" + p(x)y' + q(x)y = f(x) (1) where p(x), q(x) and f(x) are continuous functions on an interval I. The particular solution of this equations is given as: Yp(x) = u₁(x)y₁(x) + U₂(x)y2(x) (2) where y₁(x) and y2(x) are two linearly independent solutions of the corresponding homogeneous equation, and u₁ (x) and u2(x) are functions to be determined. (a) Show that the particular solution yp(x) can be expressed in terms of an integral involving the Green's function G(x, t): where G(x, t) is defined as: yp(x) = f*G(x,t)f(t) dt G(x, t) = Y1(t) y2(x) - Y₁(x) y2(t) W(t) where W(t) is the Wronskian of y₁ (t) and y2(t). (b) Verify that the integral form of yp(x) in Eq. (3) satisfies the following initial conditions: y(xo) 0, y'(xo) = 0 (c) In the light of previous parts, use the Green's function (Eq.3) to solve the following initial value problem: y" y = e²x with y(0) = 0 and y'(0) = 0 (3) (4)
Transcribed Image Text:Consider a second-order linear non-homogeneous differential equation of the form: y" + p(x)y' + q(x)y = f(x) (1) where p(x), q(x) and f(x) are continuous functions on an interval I. The particular solution of this equations is given as: Yp(x) = u₁(x)y₁(x) + U₂(x)y2(x) (2) where y₁(x) and y2(x) are two linearly independent solutions of the corresponding homogeneous equation, and u₁ (x) and u2(x) are functions to be determined. (a) Show that the particular solution yp(x) can be expressed in terms of an integral involving the Green's function G(x, t): where G(x, t) is defined as: yp(x) = f*G(x,t)f(t) dt G(x, t) = Y1(t) y2(x) - Y₁(x) y2(t) W(t) where W(t) is the Wronskian of y₁ (t) and y2(t). (b) Verify that the integral form of yp(x) in Eq. (3) satisfies the following initial conditions: y(xo) 0, y'(xo) = 0 (c) In the light of previous parts, use the Green's function (Eq.3) to solve the following initial value problem: y" y = e²x with y(0) = 0 and y'(0) = 0 (3) (4)
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