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(aE1 + bE2 + c)z(k, l) = F(k,l), (5.183) e constants, and F(k, l) satisfies the condition (aE1 + bE2 + c)F(k,l) = 0. (5.184)

(aE1 + bE2 + c)z(k, l) = F(k,l), (5.183) e constants, and F(k, l) satisfies the condition (aE1 + bE2 + c)F(k,l) = 0. (5.184)

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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(aE1 + bE2 + c)z(k, l) = F(k,l), (5.183) where a, b, and c are constants, and F(k, l) satisfies the condition (aE1 + bE2 + c)F(k,l) = 0. (5.184)
Transcribed Image Text:(aE1 + bE2 + c)z(k, l) = F(k,l), (5.183) where a, b, and c are constants, and F(k, l) satisfies the condition (aE1 + bE2 + c)F(k,l) = 0. (5.184)
5.5.5 Example E O in equation (5.183) and continue to assume that F(k, l) is a solution to the homogeneous equation. Under these conditions, we have Assume that c= (aE1 +bE2)z(k, l) = F(k,l) (5.189) and (aE1 + bE2)F(k, l) = 0. (5.190) The solution to the last equation is (F(k, e) = (-6/a)*f(e + k), (5.191) where f is an arbitrary function of l + k. Examination of the left-hand side of equation (5.189) shows that it is of a form such that Laplace's method can be used to obtain a solution. If we let k +l = m = constant, Vk = z(k, l) = z(k, m – k), (5.192) then vk satisfies the first-order inhomogeneous equation avk+1 +bvk = (-b/a)*f(m), (5.193) where we have used the results of equations (5.191) and (5.192) to replace the right-hand side of equation (5.189). Note that f(m) is a constant. Solving for Vk gives k k Uk = A (5.194) a where A is an arbitrary constant. Replacing m by l+k and A by an arbitrary function of l +k gives the complete solution to equation (5.189), under the assumption of equation (5.190), k z(k, l) = g(e + k) - (--) sce+ k). f(l + k). (5.195)
Transcribed Image Text:5.5.5 Example E O in equation (5.183) and continue to assume that F(k, l) is a solution to the homogeneous equation. Under these conditions, we have Assume that c= (aE1 +bE2)z(k, l) = F(k,l) (5.189) and (aE1 + bE2)F(k, l) = 0. (5.190) The solution to the last equation is (F(k, e) = (-6/a)*f(e + k), (5.191) where f is an arbitrary function of l + k. Examination of the left-hand side of equation (5.189) shows that it is of a form such that Laplace's method can be used to obtain a solution. If we let k +l = m = constant, Vk = z(k, l) = z(k, m – k), (5.192) then vk satisfies the first-order inhomogeneous equation avk+1 +bvk = (-b/a)*f(m), (5.193) where we have used the results of equations (5.191) and (5.192) to replace the right-hand side of equation (5.189). Note that f(m) is a constant. Solving for Vk gives k k Uk = A (5.194) a where A is an arbitrary constant. Replacing m by l+k and A by an arbitrary function of l +k gives the complete solution to equation (5.189), under the assumption of equation (5.190), k z(k, l) = g(e + k) - (--) sce+ k). f(l + k). (5.195)
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